IODS course project
Mikael
options(knitr.duplicate.label = 'allow')Chapter 1 - About the project:
IODS is a great course where one can learn a lot of useful things. It seemed difficult at first glance, but after spending some time and doing some research, one can certainly learn a lot and for me, the course has in a way, molded my perspective on data science.
The course is all about exploring the possibilities of ODS and getting familiar with those, as well as having fun.
Chapter 2 - Regression and model validation
I have created an appropriate analysis dataset and excluded unwanted observations. Here we have analysis on the dataset and it’s variables. - Describe your work and results clearly. - Assume the reader has an introductory course level understanding of writing and reading R code as well as statistical methods - Assume the reader has no previous knowledge of your data or the more advanced methods you are using
For these excercises the libraries “dplyr”, “ggplot2” and “GGally” are necessary. When reading the code these need to be installed and read. I have also included cache = F, since without it my version of R refused to knit the plots I made
- First we read the dataset I created into R and explore its contents, dimensions and structure
cache=F
library("dplyr")##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary("ggplot2")
library("GGally")##
## Attaching package: 'GGally'## The following object is masked from 'package:dplyr':
##
## nasalibrary("lattice")
Students2014 <- read.table("~/Documents/IODS-project/data/learning2014/learning2014.txt", header = TRUE, sep = " ")
print(Students2014)## gender Age Attitude deep stra surf Points
## 1 F 53 37 3.583333 3.375 2.583333 25
## 2 M 55 31 2.916667 2.750 3.166667 12
## 3 F 49 25 3.500000 3.625 2.250000 24
## 4 M 53 35 3.500000 3.125 2.250000 10
## 5 M 49 37 3.666667 3.625 2.833333 22
## 6 F 38 38 4.750000 3.625 2.416667 21
## 7 M 50 35 3.833333 2.250 1.916667 21
## 8 F 37 29 3.250000 4.000 2.833333 31
## 9 M 37 38 4.333333 4.250 2.166667 24
## 10 F 42 21 4.000000 3.500 3.000000 26
## 11 M 37 39 3.583333 3.625 2.666667 31
## 12 F 34 38 3.833333 4.750 2.416667 31
## 13 F 34 24 4.250000 3.625 2.250000 23
## 14 F 34 30 3.333333 3.500 2.750000 25
## 15 M 35 26 4.166667 1.750 2.333333 21
## 16 F 33 41 3.666667 3.875 2.333333 31
## 17 F 32 26 4.083333 1.375 2.916667 20
## 18 F 44 26 3.500000 3.250 2.500000 22
## 19 M 29 17 4.083333 3.000 3.750000 9
## 20 F 30 27 4.000000 3.750 2.750000 24
## 21 M 27 39 3.916667 2.625 2.333333 28
## 22 M 29 34 4.000000 2.375 2.416667 30
## 23 F 31 27 4.000000 3.625 3.000000 24
## 24 F 37 23 3.666667 2.750 2.416667 9
## 25 F 26 37 3.666667 1.750 2.833333 26
## 26 F 26 44 4.416667 3.250 3.166667 32
## 27 M 30 41 3.916667 4.000 3.000000 32
## 28 F 33 37 3.750000 3.625 2.000000 33
## 29 F 33 25 3.250000 2.875 3.500000 29
## 30 M 28 30 3.583333 3.000 3.750000 30
## 31 M 26 34 4.916667 1.625 2.500000 19
## 32 F 27 32 3.583333 3.250 2.083333 23
## 33 F 25 20 2.916667 3.500 2.416667 19
## 34 F 31 24 3.666667 3.000 2.583333 12
## 35 M 20 42 4.500000 3.250 1.583333 10
## 36 F 39 16 4.083333 1.875 2.833333 11
## 37 M 38 31 3.833333 4.375 1.833333 20
## 38 M 24 38 3.250000 3.625 2.416667 26
## 39 M 26 38 2.333333 2.500 3.250000 31
## 40 M 25 33 3.333333 1.250 3.416667 20
## 41 F 30 17 4.083333 4.000 3.416667 23
## 42 F 25 25 2.916667 3.000 3.166667 12
## 43 M 30 32 3.333333 2.500 3.500000 24
## 44 F 48 35 3.833333 4.875 2.666667 17
## 45 F 24 32 3.666667 5.000 2.416667 29
## 46 F 40 42 4.666667 4.375 3.583333 23
## 47 M 25 31 3.750000 3.250 2.083333 28
## 48 F 23 39 3.416667 4.000 3.750000 31
## 49 F 25 19 4.166667 3.125 2.916667 23
## 50 F 23 21 2.916667 2.500 2.916667 25
## 51 M 27 25 4.166667 3.125 2.416667 18
## 52 M 25 32 3.583333 3.250 3.000000 19
## 53 M 23 32 2.833333 2.125 3.416667 22
## 54 F 23 26 4.000000 2.750 2.916667 25
## 55 F 23 23 2.916667 2.375 3.250000 21
## 56 F 45 38 3.000000 3.125 3.250000 9
## 57 F 22 28 4.083333 4.000 2.333333 28
## 58 F 23 33 2.916667 4.000 3.250000 25
## 59 M 21 48 3.500000 2.250 2.500000 29
## 60 M 21 40 4.333333 3.250 1.750000 33
## 61 F 21 40 4.250000 3.625 2.250000 33
## 62 F 21 47 3.416667 3.625 2.083333 25
## 63 F 26 23 3.083333 2.500 2.833333 18
## 64 F 25 31 4.583333 1.875 2.833333 22
## 65 F 26 27 3.416667 2.000 2.416667 17
## 66 M 21 41 3.416667 1.875 2.250000 25
## 67 F 23 34 3.416667 4.000 2.833333 28
## 68 F 22 25 3.583333 2.875 2.250000 22
## 69 F 22 21 1.583333 3.875 1.833333 26
## 70 F 22 14 3.333333 2.500 2.916667 11
## 71 F 23 19 4.333333 2.750 2.916667 29
## 72 M 22 37 4.416667 4.500 2.083333 22
## 73 M 23 32 4.833333 3.375 2.333333 21
## 74 M 24 28 3.083333 2.625 2.416667 28
## 75 F 22 41 3.000000 4.125 2.750000 33
## 76 F 23 25 4.083333 2.625 3.250000 16
## 77 M 22 28 4.083333 2.250 1.750000 31
## 78 M 20 38 3.750000 2.750 2.583333 22
## 79 M 22 31 3.083333 3.000 3.333333 31
## 80 M 21 35 4.750000 1.625 2.833333 23
## 81 F 22 36 4.250000 1.875 2.500000 26
## 82 F 23 26 4.166667 3.375 2.416667 12
## 83 M 21 44 4.416667 3.750 2.416667 26
## 84 M 22 45 3.833333 2.125 2.583333 31
## 85 M 29 32 3.333333 2.375 3.000000 19
## 86 F 29 39 3.166667 2.750 2.000000 30
## 87 F 21 25 3.166667 3.125 3.416667 12
## 88 M 28 33 3.833333 3.500 2.833333 17
## 89 F 21 33 4.250000 2.625 2.250000 18
## 90 F 30 30 3.833333 3.375 2.750000 19
## 91 F 21 29 3.666667 2.250 3.916667 21
## 92 M 23 33 3.833333 3.000 2.333333 24
## 93 F 21 33 3.833333 4.000 2.750000 28
## 94 F 21 35 3.833333 3.500 2.750000 17
## 95 F 20 36 3.666667 2.625 2.916667 18
## 96 M 22 37 4.333333 2.500 2.083333 17
## 97 M 21 42 3.750000 3.750 3.666667 23
## 98 M 21 32 4.166667 3.625 2.833333 26
## 99 F 20 50 4.000000 4.125 3.416667 28
## 100 M 22 47 4.000000 4.375 1.583333 31
## 101 F 20 36 4.583333 2.625 2.916667 27
## 102 F 20 36 3.666667 4.000 3.000000 25
## 103 M 24 29 3.666667 2.750 2.916667 23
## 104 F 20 35 3.833333 2.750 2.666667 21
## 105 F 19 40 2.583333 1.375 3.000000 27
## 106 F 21 35 3.500000 2.250 2.750000 28
## 107 F 21 32 3.083333 3.625 3.083333 23
## 108 F 22 26 4.250000 3.750 2.500000 21
## 109 F 25 20 3.166667 4.000 2.333333 25
## 110 F 21 27 3.083333 3.125 3.000000 11
## 111 F 22 32 4.166667 3.250 3.000000 19
## 112 F 25 33 2.250000 2.125 4.000000 24
## 113 F 20 39 3.333333 2.875 3.250000 28
## 114 M 24 33 3.083333 1.500 3.500000 21
## 115 F 20 30 2.750000 2.500 3.500000 24
## 116 M 21 37 3.250000 3.250 3.833333 24
## 117 F 20 25 4.000000 3.625 2.916667 20
## 118 F 20 29 3.583333 3.875 2.166667 19
## 119 M 31 39 4.083333 3.875 1.666667 30
## 120 F 20 36 4.250000 2.375 2.083333 22
## 121 F 22 29 3.416667 3.000 2.833333 16
## 122 F 22 21 3.083333 3.375 3.416667 16
## 123 M 21 31 3.500000 2.750 3.333333 19
## 124 M 22 40 3.666667 4.500 2.583333 30
## 125 F 21 31 4.250000 2.625 2.833333 23
## 126 F 21 23 4.250000 2.750 3.333333 19
## 127 F 21 28 3.833333 3.250 3.000000 18
## 128 F 21 37 4.416667 4.125 2.583333 28
## 129 F 20 26 3.500000 3.375 2.416667 21
## 130 F 21 24 3.583333 2.750 3.583333 19
## 131 F 25 30 3.666667 4.125 2.083333 27
## 132 M 21 28 2.083333 3.250 4.333333 24
## 133 F 24 29 4.250000 2.875 2.666667 21
## 134 F 20 24 3.583333 2.875 3.000000 20
## 135 M 21 31 4.000000 2.375 2.666667 28
## 136 F 20 19 3.333333 3.875 2.166667 12
## 137 F 20 20 3.500000 2.125 2.666667 21
## 138 F 18 38 3.166667 4.000 2.250000 28
## 139 F 21 34 3.583333 3.250 2.666667 31
## 140 F 19 37 3.416667 2.625 3.333333 18
## 141 F 21 29 4.250000 2.750 3.500000 25
## 142 F 20 23 3.250000 4.000 2.750000 19
## 143 M 21 41 4.416667 3.000 2.000000 21
## 144 F 20 27 3.250000 3.375 2.833333 16
## 145 F 21 35 3.916667 3.875 3.500000 7
## 146 F 20 34 3.583333 3.250 2.500000 21
## 147 F 18 32 4.500000 3.375 3.166667 17
## 148 M 22 33 3.583333 4.125 3.083333 22
## 149 F 22 33 3.666667 3.500 2.916667 18
## 150 M 24 35 2.583333 2.000 3.166667 25
## 151 F 19 32 4.166667 3.625 2.500000 24
## 152 F 20 31 3.250000 3.375 3.833333 23
## 153 F 20 28 4.333333 2.125 2.250000 23
## 154 F 17 17 3.916667 4.625 3.416667 26
## 155 M 19 19 2.666667 2.500 3.750000 12
## 156 F 20 35 3.083333 2.875 3.000000 32
## 157 F 20 24 3.750000 2.750 2.583333 22
## 158 F 20 21 4.166667 4.000 3.333333 20
## 159 F 20 29 4.166667 2.375 2.833333 21
## 160 F 19 19 3.250000 3.875 3.000000 23
## 161 F 19 20 4.083333 3.375 2.833333 20
## 162 F 22 42 2.916667 1.750 3.166667 28
## 163 M 35 41 3.833333 3.000 2.750000 31
## 164 F 18 37 3.166667 2.625 3.416667 18
## 165 F 19 36 3.416667 2.625 3.000000 30
## 166 M 21 18 4.083333 3.375 2.666667 19#2.
dim(Students2014)## [1] 166 7str(Students2014)## 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ Age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ Attitude: int 37 31 25 35 37 38 35 29 38 21 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ Points : int 25 12 24 10 22 21 21 31 24 26 ...Second we display graphical implementation of the data
Overview:
plot(Students2014)
Results without variable ‘gender’: pairs(Students2014[-1])
pairs(Students2014[-1])
Last, a ggpairs-graphic for possibly a more clear display:
ggpairs(Students2014, mapping = aes(col = gender, alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)))
Next we have summaries for the included variables and scatterplots to clarify their influence on the ‘points’ -variable:
library(ggplot2)
qplot(Attitude, Points, data = Students2014) + geom_smooth(method = "lm")
qplot(Age, Points, data = Students2014) + geom_smooth(method = "lm")
qplot(gender, Points, data = Students2014) + geom_smooth(method = "lm")
qplot(deep, Points, data = Students2014) + geom_smooth(method = "lm")
qplot(stra, Points, data = Students2014) + geom_smooth(method = "lm")
qplot(surf, Points, data = Students2014) + geom_smooth(method = "lm")
Next some more illustration of the effects of the other variables to the ‘points’-variable and summaries of these
- In the following simple regression models Points is used as the target variable while other variables are used as explanatory variables. There is also a summary provided for all of the individual regressions.
M_Attitude <- lm(Points ~ Attitude, data = Students2014)
summary(M_Attitude)##
## Call:
## lm(formula = Points ~ Attitude, data = Students2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.9763 -3.2119 0.4339 4.1534 10.6645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.63715 1.83035 6.358 1.95e-09 ***
## Attitude 0.35255 0.05674 6.214 4.12e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1856
## F-statistic: 38.61 on 1 and 164 DF, p-value: 4.119e-09M_Age <- lm(Points ~ Age, data = Students2014)
summary(M_Age)##
## Call:
## lm(formula = Points ~ Age, data = Students2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.0360 -3.7531 0.0958 4.6762 10.8128
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 24.52150 1.57339 15.585 <2e-16 ***
## Age -0.07074 0.05901 -1.199 0.232
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.887 on 164 degrees of freedom
## Multiple R-squared: 0.008684, Adjusted R-squared: 0.00264
## F-statistic: 1.437 on 1 and 164 DF, p-value: 0.2324M_gender <- lm(Points ~ gender, data = Students2014)
summary(M_gender)##
## Call:
## lm(formula = Points ~ gender, data = Students2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.3273 -3.3273 0.5179 4.5179 10.6727
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 22.3273 0.5613 39.776 <2e-16 ***
## genderM 1.1549 0.9664 1.195 0.234
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.887 on 164 degrees of freedom
## Multiple R-squared: 0.008632, Adjusted R-squared: 0.002587
## F-statistic: 1.428 on 1 and 164 DF, p-value: 0.2338M_deep <- lm(Points ~ deep, data = Students2014)
summary(M_deep)##
## Call:
## lm(formula = Points ~ deep, data = Students2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.6913 -3.6935 0.2862 4.9957 10.3537
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 23.1141 3.0908 7.478 4.31e-12 ***
## deep -0.1080 0.8306 -0.130 0.897
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.913 on 164 degrees of freedom
## Multiple R-squared: 0.000103, Adjusted R-squared: -0.005994
## F-statistic: 0.01689 on 1 and 164 DF, p-value: 0.8967M_stra <- lm(Points ~ stra, data = Students2014)
summary(M_stra)##
## Call:
## lm(formula = Points ~ stra, data = Students2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.5581 -3.8198 0.1042 4.3024 10.1394
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 19.233 1.897 10.141 <2e-16 ***
## stra 1.116 0.590 1.892 0.0603 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.849 on 164 degrees of freedom
## Multiple R-squared: 0.02135, Adjusted R-squared: 0.01538
## F-statistic: 3.578 on 1 and 164 DF, p-value: 0.06031M_surf <- lm(Points ~ surf, data = Students2014)
summary(M_surf)##
## Call:
## lm(formula = Points ~ surf, data = Students2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.6539 -3.3744 0.3574 4.4734 10.2234
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 27.2017 2.4432 11.134 <2e-16 ***
## surf -1.6091 0.8613 -1.868 0.0635 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.851 on 164 degrees of freedom
## Multiple R-squared: 0.02084, Adjusted R-squared: 0.01487
## F-statistic: 3.49 on 1 and 164 DF, p-value: 0.06351Looking at the output of the summaries, the most trustworthy explanatory variables are Attitude, Age, surf and stra. These are confirmed by the scatterplots of the individual dependences of points on the other variables. Age, Attitude and stra can be seen to probably have influence on points. The values of the regression make these assumption reasonable, with a quite low margin for failure. (Failure in this case means selecting output or in other words a random sample where the assumptions fail to reflect or descibe the truth/dependence of the variables).
The mean, standard error and variance for the dataset’s variables:
sapply(Students2014, mean, na.rm = TRUE)## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA## gender Age Attitude deep stra surf Points
## NA 25.512048 31.427711 3.679719 3.121235 2.787149 22.716867sapply(Students2014, sd, na.rm = TRUE)## Warning in var(if (is.vector(x) || is.factor(x)) x else as.double(x), na.rm = na.rm): Calling var(x) on a factor x is deprecated and will become an error.
## Use something like 'all(duplicated(x)[-1L])' to test for a constant vector.## gender Age Attitude deep stra surf Points
## 0.4742358 7.7660785 7.2990794 0.5541369 0.7718318 0.5288405 5.8948836sapply(Students2014, var, na.rm = TRUE)## Warning in FUN(X[[i]], ...): Calling var(x) on a factor x is deprecated and will become an error.
## Use something like 'all(duplicated(x)[-1L])' to test for a constant vector.## gender Age Attitude deep stra surf
## 0.2248996 60.3119752 53.2765608 0.3070677 0.5957244 0.2796722
## Points
## 34.7496532Below is a regression model where exam points is the target/dependent variable, with three explanatory variables. These explanatory variables were chosen because it can be seen that they correlate with the variable that we are attempting to explain (points). At the same time the model is drawn in several different ways to help interpret and understand it’s relevance.
Model3 <- lm(formula = Points ~ Attitude + Age + stra, data = Students2014)
plot(Model3)
The following functions work in my R-project, but for some reason they refused to knit to HTML. I have taken measures to enable the knitting of error terms etc, and it still won’t work. (I used these functions to make some interesting plots that I discuss later in the exercise. These were not individually necessary for ch2, and I have provided a collection of the mentioned plots in an other way also. This collection is included in the code). Still, I wanted to post them here to show what I did in another way. Readers please note: These were not as such demanded, and are provided in the necessary form later. If experimentation is done on them, I recommend copying them to an R-document, along with all the other necessary elements
r.squared(Model3, model = NULL, type = c(“Attitude”, “Age”, “stra”), dfcor = TRUE) #Normal Q-Q r.squared(Model3, model = NULL, type = c(“Attitude”, “Age”, “stra”), dfcor = FALSE) r.squared(Model3, model = “lm”, type = c(“Attitude”, “Age”, “stra”), dfcor = TRUE) #Res vs Lev r.squared(Model3, model = “lm”, type = c(“Attitude”, “Age”, “stra”), dfcor = FALSE) #Res vs Fit
- Finally, a summary of the model:
summary(Model3)##
## Call:
## lm(formula = Points ~ Attitude + Age + stra, data = Students2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18.1149 -3.2003 0.3303 3.4129 10.7599
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.89543 2.64834 4.114 6.17e-05 ***
## Attitude 0.34808 0.05622 6.191 4.72e-09 ***
## Age -0.08822 0.05302 -1.664 0.0981 .
## stra 1.00371 0.53434 1.878 0.0621 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.26 on 162 degrees of freedom
## Multiple R-squared: 0.2182, Adjusted R-squared: 0.2037
## F-statistic: 15.07 on 3 and 162 DF, p-value: 1.07e-08The summary provides seemingly significant results, with considerably high correlation to points of all the other variables. The error has remained quite low in relation to the amount of variables used. The r-squared shows generally how close the data on the variables is to the regression line. Experimenting with different inputs to the function r-squared 4 different but interesting outputs can be found with this ampunt of experimentation. It seems that the Multiple R^2 is quite low, so it’s not too significant (It doesn’t deny the validity of the regression. The variants of the r-squared functions produced the Residuals vs Fitted, Residuals vs Leverage, and Normal Q-Q plots, as well as Scale location.
- Next, three diagnostic plots are combined to help consider the validity of the model
par(mfrow = c(2,2))
plot(Model3, which = c(1,2,5))
Linear regression models have a few general assumtions: 1. Linearity 2. The errors of the model are normally distributed. 2. The errors are not correlated. 3. The sizes of the errors do not depend on the variables used to explain the target variable.
Now let’s think how the plots produced correspond (or not) to these assumption:
The Q–Q-plot demonstrates how the standardised residuals of the model fit to the theory or reasoning behind the model. Therefore the normal distribution assumption seems to be true for the model.
The residuals vs. fitted values -plot does not seem to be regular/subjected to any pattern, meaning that the errors are not correlated to the explanatory variables and their size is independent.
Therefore all of the assumptions are valid for the model created.
Chapter 3 - Logistic Regression
There was a problem reading the file into R from my computer; The code worked but the data didn’t print out correctly, even though the same exact code combinations worked elsewhere, so I have included the necessary parts of the wrangling part to create the dataset again.
cache=F
library(tidyr)
library(dplyr)
library(ggplot2)
library(gmodels)student_mat <- read.csv("~/Documents/IODS-project/data/student-mat.csv", sep = ";")
student_por <- read.csv("~/Documents/IODS-project/data/student-por.csv", sep = ";")
join_by <- c("school","sex","age","address","famsize","Pstatus","Medu","Fedu","Mjob","Fjob","reason","nursery","internet")
math_por <- inner_join(student_mat, student_por, by = join_by, suffix = c(".student_mat", ".student_por"))
alc <- select(math_por, one_of(join_by))
notjoined_columns <- colnames(student_mat)[!colnames(student_mat) %in% join_by]
# for every column name not used for joining...
for(column_name in notjoined_columns) {
# select two columns from 'math_por' with the same original name
two_columns <- select(math_por, starts_with(column_name))
# select the first column vector of those two columns
first_column <- select(two_columns, 1)[[1]]
# if that first column vector is numeric...
if(is.numeric(first_column)) {
# take a rounded average of each row of the two columns and
# add the resulting vector to the alc data frame
alc[column_name] <- round(rowMeans(two_columns))
} else { # else if it's not numeric...
# add the first column vector to the alc data frame
alc[column_name] <- first_column
}
}
alc <- mutate(alc, alc_use = (Dalc + Walc) / 2)
alc <- mutate(alc, high_use = alc_use > 2)- Let’s read the alc data. (This works but the original output wasn’t maintained from when I first wrote the code, as I explained)
#alc <- read.csv("~/Documents/IODS-project/data/create_alc.R", row.names = NULL)
names(alc)## [1] "school" "sex" "age" "address" "famsize"
## [6] "Pstatus" "Medu" "Fedu" "Mjob" "Fjob"
## [11] "reason" "nursery" "internet" "guardian" "traveltime"
## [16] "studytime" "failures" "schoolsup" "famsup" "paid"
## [21] "activities" "higher" "romantic" "famrel" "freetime"
## [26] "goout" "Dalc" "Walc" "health" "absences"
## [31] "G1" "G2" "G3" "alc_use" "high_use"For my variables, I chose sex, final exam scores, famsize and studytime. My hypotheses are as follows: Fewer females than males who took part consume a lot of alcohol. I would also assume that a person who doesn’t consume a lot of alcohol gets better grades than someone who drinks more.
Now we must test the hypothesis.
alc %>% group_by(sex, high_use) %>%summarise(count=n(),mean_grade=mean(G3))## # A tibble: 4 x 4
## # Groups: sex [?]
## sex high_use count mean_grade
## <fct> <lgl> <int> <dbl>
## 1 F FALSE 156 11.4
## 2 F TRUE 42 11.7
## 3 M FALSE 112 12.2
## 4 M TRUE 72 10.3First we have the crosstabulation for alcohol usage, sex and the average of the final exam grades. The results of this tabulation tell us two things: i.) A higher percentage of men than women use alcohol a lot. ii.) Interestingly, those women who drink more have slightly higher grades. In contrast, binge-drinking men have worse grades than those who don’t drink a lot.
Then let’s create a boxplot of high alcohol usages effects to the final grades.
g1 <- ggplot(alc, aes(x= high_use, y=G3), col=sex)
g1 + geom_boxplot()+ylab("grade")
It is clear that on average, alcohol-lovers have worse grades than moderate drinkers.
Then we have a barplot describing the relationship; alcohol usage vs. study time. Those who do not drink much are included in the darkred bar and those who drink a lot in the darkblue bar.
counts <- table(alc$high_use, alc$studytime)
barplot(counts, main = "Usage of alcohol", xlab = "studytime", col = c("darkred", "darkblue"), beside = TRUE)
The key ratio described is the share of the drinkaholics in each category of study time. The shares are about equally small in 4 and 3 but get bigger in 2 and in 1 it is almost as common to drink a lot than not to drink a lot.
Then we compare alcohol usage to family size.
counts2 <- table(alc$high_use, alc$famsize)
barplot(counts2, main = "Usage of alcohol", xlab = "famsize", col = c("black", "gold"), beside = TRUE)
The result is that students from small families, are somewhat more likely to drink a lot than those from big families.
- The logistic regression describing our varibles follows.
m <- glm(high_use ~ famsize + sex + G3 + studytime, data = alc, family = "binomial")
summary(m)##
## Call:
## glm(formula = high_use ~ famsize + sex + G3 + studytime, family = "binomial",
## data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.4556 -0.8448 -0.6728 1.1674 2.1425
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.36140 0.50827 0.711 0.47706
## famsizeLE3 0.31601 0.25722 1.229 0.21923
## sexM 0.67284 0.24445 2.752 0.00592 **
## G3 -0.07463 0.03598 -2.074 0.03806 *
## studytime -0.41808 0.15933 -2.624 0.00869 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 465.68 on 381 degrees of freedom
## Residual deviance: 435.28 on 377 degrees of freedom
## AIC: 445.28
##
## Number of Fisher Scoring iterations: 4Coefficients as odds ratios:
OR <- coef(m) %>% exp
OR## (Intercept) famsizeLE3 sexM G3 studytime
## 1.4353404 1.3716456 1.9598015 0.9280889 0.6583124COnfidence intervals:
CI <- confint(m) %>% exp## Waiting for profiling to be done...CI## 2.5 % 97.5 %
## (Intercept) 0.5317356 3.9213022
## famsizeLE3 0.8244681 2.2647911
## sexM 1.2165115 3.1774177
## G3 0.8641265 0.9954148
## studytime 0.4773416 0.8930376- Sex ja study time seem to have a statistical relationship to alcohol use. That’s why we modify the model to include only sex and study time.
m <- glm(high_use ~ sex + studytime, data = alc, family = "binomial")
probabilities <- predict(m, type = "response")
alc <- mutate(alc, probability = probabilities)
alc <- mutate(alc, prediction = probability > 0.3)
select(alc, studytime, sex, high_use, probability, prediction) %>% tail(10)## studytime sex high_use probability prediction
## 373 1 M FALSE 0.4795667 TRUE
## 374 1 M TRUE 0.4795667 TRUE
## 375 3 F FALSE 0.1536065 FALSE
## 376 1 F FALSE 0.3222140 TRUE
## 377 3 F FALSE 0.1536065 FALSE
## 378 2 F FALSE 0.2270396 FALSE
## 379 2 F FALSE 0.2270396 FALSE
## 380 2 F FALSE 0.2270396 FALSE
## 381 1 M TRUE 0.4795667 TRUE
## 382 1 M TRUE 0.4795667 TRUEtable(high_use = alc$high_use, prediction = alc$prediction)## prediction
## high_use FALSE TRUE
## FALSE 158 110
## TRUE 37 77CrossTable(alc$high_use, alc$prediction)##
##
## Cell Contents
## |-------------------------|
## | N |
## | Chi-square contribution |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 382
##
##
## | alc$prediction
## alc$high_use | FALSE | TRUE | Row Total |
## -------------|-----------|-----------|-----------|
## FALSE | 158 | 110 | 268 |
## | 3.283 | 3.424 | |
## | 0.590 | 0.410 | 0.702 |
## | 0.810 | 0.588 | |
## | 0.414 | 0.288 | |
## -------------|-----------|-----------|-----------|
## TRUE | 37 | 77 | 114 |
## | 7.719 | 8.049 | |
## | 0.325 | 0.675 | 0.298 |
## | 0.190 | 0.412 | |
## | 0.097 | 0.202 | |
## -------------|-----------|-----------|-----------|
## Column Total | 195 | 187 | 382 |
## | 0.510 | 0.490 | |
## -------------|-----------|-----------|-----------|
##
## counts3 <- table(alc$high_use, alc$prediction)
barplot(counts3, main = "Usage of alcohol", xlab = "high_use", col = c("pink", "purple"), beside = TRUE)
g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
g + geom_point()
table(high_use = alc$high_use, prediction = alc$prediction) %>% prop.table %>% addmargins## prediction
## high_use FALSE TRUE Sum
## FALSE 0.41361257 0.28795812 0.70157068
## TRUE 0.09685864 0.20157068 0.29842932
## Sum 0.51047120 0.48952880 1.00000000loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
loss_func(class = alc$high_use, prob = 0)## [1] 0.2984293According to the results, the model provided provides a sense of reality, even if it it statistically not very reliable. By simply guessing what the results would be, the ideas that one comes up with could be similar but more accurate. What the model probably works for though, is that is offers some slight validation for any guesses made. Not much can be concluded from the model alone however, since the ratio of wrong predictions is quite large.
- Here is ten-fold cross-validation of the model produced earlier. The average prediction error is higher than the one on datacamp. Let us see if one with a lower error can be found in Q8.
library(boot)##
## Attaching package: 'boot'## The following object is masked from 'package:lattice':
##
## melanomacv <- cv.glm(data = alc, cost = loss_func, glmfit = m, K = 10)
cv$delta[1]## [1] 0.2984293- Let us make and compare some different models. First there is a model that is slightly larger than the original one. Second, there is one with a lot more predictors. Next a smaller model with different variables than the original m.
n <- glm(high_use ~ sex + studytime + famsize + failures + activities + absences, data = alc, family = "binomial")
probabilities <- predict(n, type = "response")
select(alc, studytime, sex, high_use, probability, prediction) %>% tail(10)## studytime sex high_use probability prediction
## 373 1 M FALSE 0.4795667 TRUE
## 374 1 M TRUE 0.4795667 TRUE
## 375 3 F FALSE 0.1536065 FALSE
## 376 1 F FALSE 0.3222140 TRUE
## 377 3 F FALSE 0.1536065 FALSE
## 378 2 F FALSE 0.2270396 FALSE
## 379 2 F FALSE 0.2270396 FALSE
## 380 2 F FALSE 0.2270396 FALSE
## 381 1 M TRUE 0.4795667 TRUE
## 382 1 M TRUE 0.4795667 TRUEtable(high_use = alc$high_use, prediction = alc$prediction)## prediction
## high_use FALSE TRUE
## FALSE 158 110
## TRUE 37 77CrossTable(alc$high_use, alc$prediction)##
##
## Cell Contents
## |-------------------------|
## | N |
## | Chi-square contribution |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 382
##
##
## | alc$prediction
## alc$high_use | FALSE | TRUE | Row Total |
## -------------|-----------|-----------|-----------|
## FALSE | 158 | 110 | 268 |
## | 3.283 | 3.424 | |
## | 0.590 | 0.410 | 0.702 |
## | 0.810 | 0.588 | |
## | 0.414 | 0.288 | |
## -------------|-----------|-----------|-----------|
## TRUE | 37 | 77 | 114 |
## | 7.719 | 8.049 | |
## | 0.325 | 0.675 | 0.298 |
## | 0.190 | 0.412 | |
## | 0.097 | 0.202 | |
## -------------|-----------|-----------|-----------|
## Column Total | 195 | 187 | 382 |
## | 0.510 | 0.490 | |
## -------------|-----------|-----------|-----------|
##
## g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
g + geom_point()
table(high_use = alc$high_use, prediction = alc$prediction) %>% prop.table %>% addmargins## prediction
## high_use FALSE TRUE Sum
## FALSE 0.41361257 0.28795812 0.70157068
## TRUE 0.09685864 0.20157068 0.29842932
## Sum 0.51047120 0.48952880 1.00000000loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
loss_func(class = alc$high_use, prob = 0)## [1] 0.2984293cv <- cv.glm(data = alc, cost = loss_func, glmfit = m, K = 10)
cv$delta[1]## [1] 0.3062827o <- glm(high_use ~ sex + studytime + famsize + failures + activities + absences + health + goout + freetime + romantic, data = alc, family = "binomial")
probabilities <- predict(o, type = "response")
select(alc, studytime, sex, high_use, probability, prediction) %>% tail(10)## studytime sex high_use probability prediction
## 373 1 M FALSE 0.4795667 TRUE
## 374 1 M TRUE 0.4795667 TRUE
## 375 3 F FALSE 0.1536065 FALSE
## 376 1 F FALSE 0.3222140 TRUE
## 377 3 F FALSE 0.1536065 FALSE
## 378 2 F FALSE 0.2270396 FALSE
## 379 2 F FALSE 0.2270396 FALSE
## 380 2 F FALSE 0.2270396 FALSE
## 381 1 M TRUE 0.4795667 TRUE
## 382 1 M TRUE 0.4795667 TRUEtable(high_use = alc$high_use, prediction = alc$prediction)## prediction
## high_use FALSE TRUE
## FALSE 158 110
## TRUE 37 77CrossTable(alc$high_use, alc$prediction)##
##
## Cell Contents
## |-------------------------|
## | N |
## | Chi-square contribution |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 382
##
##
## | alc$prediction
## alc$high_use | FALSE | TRUE | Row Total |
## -------------|-----------|-----------|-----------|
## FALSE | 158 | 110 | 268 |
## | 3.283 | 3.424 | |
## | 0.590 | 0.410 | 0.702 |
## | 0.810 | 0.588 | |
## | 0.414 | 0.288 | |
## -------------|-----------|-----------|-----------|
## TRUE | 37 | 77 | 114 |
## | 7.719 | 8.049 | |
## | 0.325 | 0.675 | 0.298 |
## | 0.190 | 0.412 | |
## | 0.097 | 0.202 | |
## -------------|-----------|-----------|-----------|
## Column Total | 195 | 187 | 382 |
## | 0.510 | 0.490 | |
## -------------|-----------|-----------|-----------|
##
## g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
g + geom_point()
table(high_use = alc$high_use, prediction = alc$prediction) %>% prop.table %>% addmargins## prediction
## high_use FALSE TRUE Sum
## FALSE 0.41361257 0.28795812 0.70157068
## TRUE 0.09685864 0.20157068 0.29842932
## Sum 0.51047120 0.48952880 1.00000000loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
loss_func(class = alc$high_use, prob = 0)## [1] 0.2984293cv <- cv.glm(data = alc, cost = loss_func, glmfit = m, K = 10)
cv$delta[1]## [1] 0.3115183p <- glm(high_use ~ failures + activities + absences + health + goout + freetime + romantic, data = alc, family = "binomial")
probabilities <- predict(p, type = "response")
select(alc, studytime, sex, high_use, probability, prediction) %>% tail(10)## studytime sex high_use probability prediction
## 373 1 M FALSE 0.4795667 TRUE
## 374 1 M TRUE 0.4795667 TRUE
## 375 3 F FALSE 0.1536065 FALSE
## 376 1 F FALSE 0.3222140 TRUE
## 377 3 F FALSE 0.1536065 FALSE
## 378 2 F FALSE 0.2270396 FALSE
## 379 2 F FALSE 0.2270396 FALSE
## 380 2 F FALSE 0.2270396 FALSE
## 381 1 M TRUE 0.4795667 TRUE
## 382 1 M TRUE 0.4795667 TRUEtable(high_use = alc$high_use, prediction = alc$prediction)## prediction
## high_use FALSE TRUE
## FALSE 158 110
## TRUE 37 77CrossTable(alc$high_use, alc$prediction)##
##
## Cell Contents
## |-------------------------|
## | N |
## | Chi-square contribution |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 382
##
##
## | alc$prediction
## alc$high_use | FALSE | TRUE | Row Total |
## -------------|-----------|-----------|-----------|
## FALSE | 158 | 110 | 268 |
## | 3.283 | 3.424 | |
## | 0.590 | 0.410 | 0.702 |
## | 0.810 | 0.588 | |
## | 0.414 | 0.288 | |
## -------------|-----------|-----------|-----------|
## TRUE | 37 | 77 | 114 |
## | 7.719 | 8.049 | |
## | 0.325 | 0.675 | 0.298 |
## | 0.190 | 0.412 | |
## | 0.097 | 0.202 | |
## -------------|-----------|-----------|-----------|
## Column Total | 195 | 187 | 382 |
## | 0.510 | 0.490 | |
## -------------|-----------|-----------|-----------|
##
## g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
g + geom_point()
table(high_use = alc$high_use, prediction = alc$prediction) %>% prop.table %>% addmargins## prediction
## high_use FALSE TRUE Sum
## FALSE 0.41361257 0.28795812 0.70157068
## TRUE 0.09685864 0.20157068 0.29842932
## Sum 0.51047120 0.48952880 1.00000000loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
loss_func(class = alc$high_use, prob = 0)## [1] 0.2984293cv <- cv.glm(data = alc, cost = loss_func, glmfit = m, K = 10)
cv$delta[1]## [1] 0.3089005As one could expect, increasing the amount of predictors decreases the average error, assuming the predictions from the smaller model are maintained. Choosing a few optimal predictors should yield a smaller number than choosing a large number of not-so-good predictors. None of the models I created produced a smaller average error than the one on datacamp, and I would guess it to be quite difficult to find such a model. What surprised me was the amount by which the average prediction error changes between the models produced. I expected it to variate more.
summary(cars)## speed dist
## Min. : 4.0 Min. : 2.00
## 1st Qu.:12.0 1st Qu.: 26.00
## Median :15.0 Median : 36.00
## Mean :15.4 Mean : 42.98
## 3rd Qu.:19.0 3rd Qu.: 56.00
## Max. :25.0 Max. :120.00
Chapter 4 - Clustering and classification
First we load the “Boston” data from MASS package and explore it.
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectdata("Boston")
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00names(Boston)## [1] "crim" "zn" "indus" "chas" "nox" "rm" "age"
## [8] "dis" "rad" "tax" "ptratio" "black" "lstat" "medv"The “Boston”-data is comprised of 506 observations of 14 different variables. The “Names()”-function provides the names of the variables included. The function “Summary()” produces the distributions of each variable.
Now we explore the relationship between variables.
library(corrplot)## corrplot 0.84 loadedlibrary(tidyverse)## ── Attaching packages ────────────────────────────────────────────────────────────────────────────────────── tidyverse 1.2.1 ──## ✔ tibble 1.4.2 ✔ purrr 0.2.5
## ✔ readr 1.2.1 ✔ stringr 1.3.1
## ✔ tibble 1.4.2 ✔ forcats 0.3.0## ── Conflicts ───────────────────────────────────────────────────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ✖ MASS::select() masks dplyr::select()cor_matrix<-cor(Boston) %>% round(digits=2)
corrplot(cor_matrix, type="upper", cl.pos = "b",tl.pos = "d",tl.cex = 0.6 )
In the graph, the blue colour represents positive correlation between variables while the red rpresents the negatives. The darker the colour and the bigger the ball, the stronger the correlation is. For example, the big blue ball between “rad” and “tax” tells that there is a strong correlation between access to highways and property tax rate.
Now we’ll standardise the data
boston_scaled <- scale(Boston)
summary(boston_scaled)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865Now all the observations are using the same scale. Making assumptions about the summary for example is meaningful.
Then we must turn “boston_scaled” back into a data.frame
boston_scaled <- as.data.frame(boston_scaled)These lines give us the cathegorical variable “crime”
bins <- quantile(boston_scaled$crim)
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))And these replace the old variable “crim” with the cathegorical crime
boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled <- data.frame(boston_scaled, crime)Then we divide the data to train with 80% of the data and test with 20% of the data.
n <- nrow(boston_scaled)
ind <- sample(n, size = n * 0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]- Because the data has now been cathegorised into separate sets for training and testing, we can use the training-part for linear discriminant analysis, where crime rate will be predicted by all of the other variables.
lda.fit <- lda(crime ~ ., data = train)
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
classes <- as.numeric(train$crime)
plot(lda.fit, dimen = 2, col=classes)
lda.arrows(lda.fit, myscale = 2)
Looking at the bi-plot, it’s clear that the “rad”-variable acts (on its own) as a predictor of “high crime rate” in the Boston data. The other 12 variables are associated with low, medium low and medium high rates of crime. The grouping based on these 12 variables is quite vague and it is difficult to see whether any of the variables can accurately/adequately sort the associated observations.
crime_cat<-test$crime
test<-dplyr::select(test, -crime)
lda.pred<-predict(lda.fit, newdata = test)
table(correct = crime_cat, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 12 12 1 0
## med_low 3 14 5 0
## med_high 1 7 16 2
## high 0 0 0 29The amount of correct cases and of predicted cases for each of the cathegories (low, med_low, med_high, high) varies between every sample matrix. The change is to be expected as the sets we have created have been randomly classified. There is much less variation between predictions for the “high” class than there is in the predictions of the others.
- Before we practice K-means clustering, we should reload the Boston data, scale it, and afterwards measure the distances between the observations.
data(Boston)
boston_scaled1<-as.data.frame(scale(Boston))
dist_eu<-dist(boston_scaled1)
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970head(boston_scaled1)## crim zn indus chas nox rm
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916
## age dis rad tax ptratio black
## 1 -0.1198948 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159
## 2 0.3668034 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159
## 3 -0.2655490 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351
## 4 -0.8090878 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514
## 5 -0.5106743 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159
## 6 -0.3508100 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651
## lstat medv
## 1 -1.0744990 0.1595278
## 2 -0.4919525 -0.1014239
## 3 -1.2075324 1.3229375
## 4 -1.3601708 1.1815886
## 5 -1.0254866 1.4860323
## 6 -1.0422909 0.6705582The scaled Boston data will now be used for K-means clustering. It isn’t trivial (in many cases) to investigate on the number of clusters that can classify the data. Therefore, we need to first randomize the usage of a certain number of clusters.
First let’s start with a random number cluster. Let us choose k=4 and apply k-means on the data.
kmm = kmeans(boston_scaled1,6,nstart = 50 ,iter.max = 15) The elbow method is one good technique using which we can estimate the number of clusters.
library(ggplot2)
set.seed(1234)
k.max <- 15
data <- boston_scaled1
wss <- sapply(1:k.max,
function(k){kmeans(data, k)$tot.withinss})
qplot(1:k.max, wss, geom = c("point", "line"), span = 0.2,
xlab="Number of clusters K",
ylab="Total within-clusters sum of squares")## Warning: Ignoring unknown parameters: span
## Warning: Ignoring unknown parameters: span
The elbow plot seems to indicate that we may not find more than two clear clusters but it’s good to confirm predictions using another method because there is no shortage of methods for analyses like these. Let’s try the “NbClust”- package.
library(NbClust)
nb <- NbClust(boston_scaled1, diss=NULL, distance = "euclidean",
min.nc=2, max.nc=5, method = "kmeans",
index = "all", alphaBeale = 0.1)
## *** : The Hubert index is a graphical method of determining the number of clusters.
## In the plot of Hubert index, we seek a significant knee that corresponds to a
## significant increase of the value of the measure i.e the significant peak in Hubert
## index second differences plot.
## 
## *** : The D index is a graphical method of determining the number of clusters.
## In the plot of D index, we seek a significant knee (the significant peak in Dindex
## second differences plot) that corresponds to a significant increase of the value of
## the measure.
##
## *******************************************************************
## * Among all indices:
## * 12 proposed 2 as the best number of clusters
## * 6 proposed 3 as the best number of clusters
## * 3 proposed 4 as the best number of clusters
## * 3 proposed 5 as the best number of clusters
##
## ***** Conclusion *****
##
## * According to the majority rule, the best number of clusters is 2
##
##
## *******************************************************************hist(nb$Best.nc[1,], breaks = max(na.omit(nb$Best.nc[1,])))
Now, it’s easier to see that the data is described better with two clusters. With that, we should run the k-means algorithm again.
km_final = kmeans(boston_scaled1, centers = 2)
pairs(boston_scaled1[3:9], col=km_final$cluster)
The clusters in the above plot are divided into two groups and outlined using the colors red and black. Some of the pairs are better grouped than other ones in the plot. One of the important observations can be made with the “chas”-variable where the observations in all of the pairs formed by it are wrongly clustered. Still, clusters formed by the “rad” variable are better separated.
Bonus:
Now, we will use a randomly selected cluster number (k=6) and perform LDA. We shall follow the the basic steps of scaling and distance calculation. Afterwards, we will find out how the biplot looks (of the whole data set) as we try to group it into six different categories.
boston_scaled2<-as.data.frame(scale(Boston))
head(boston_scaled2)## crim zn indus chas nox rm
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916
## age dis rad tax ptratio black
## 1 -0.1198948 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159
## 2 0.3668034 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159
## 3 -0.2655490 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351
## 4 -0.8090878 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514
## 5 -0.5106743 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159
## 6 -0.3508100 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651
## lstat medv
## 1 -1.0744990 0.1595278
## 2 -0.4919525 -0.1014239
## 3 -1.2075324 1.3229375
## 4 -1.3601708 1.1815886
## 5 -1.0254866 1.4860323
## 6 -1.0422909 0.6705582set.seed(1234)
km_bs2<-kmeans(dist_eu, centers = 6)
head(km_bs2)## $cluster
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
## 6 6 6 6 6 6 6 4 4 6 4 6 6 4 4 4 6 4
## 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
## 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
## 4 6 6 1 1 6 6 6 6 6 6 4 4 6 6 6 6 6
## 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
## 1 1 1 1 6 6 6 6 6 6 1 1 1 6 6 6 6 6
## 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
## 6 6 6 6 4 4 4 6 6 6 6 6 6 6 6 6 6 6
## 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
## 6 6 6 6 4 6 6 3 3 6 4 4 4 4 4 4 4 4
## 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
## 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 4 4
## 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
## 2 2 2 2 4 4 4 4 2 2 2 2 2 2 2 2 5 2
## 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162
## 5 5 5 5 5 2 2 2 5 2 5 5 5 3 4 2 3 3
## 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
## 3 3 4 4 3 4 4 4 4 4 4 4 4 6 6 6 6 6
## 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198
## 3 6 6 6 4 6 3 6 6 1 1 1 1 1 1 1 1 1
## 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
## 1 1 1 1 1 1 1 6 6 4 3 3 3 3 3 6 4 6
## 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234
## 3 4 3 3 3 3 3 6 3 3 3 6 3 6 4 6 3 3
## 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
## 3 4 3 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
## 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270
## 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3
## 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288
## 6 6 6 3 3 6 3 3 6 6 3 6 3 1 1 1 1 6
## 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306
## 6 1 1 1 1 6 6 6 6 6 1 1 1 6 6 6 6 6
## 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324
## 6 6 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 4
## 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342
## 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 1
## 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
## 6 6 1 6 6 1 1 1 6 1 1 1 1 1 5 5 5 2
## 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
## 2 2 2 5 5 5 2 5 5 5 5 2 5 5 5 2 2 2
## 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396
## 2 2 5 2 2 2 5 5 5 5 5 2 2 2 2 2 2 2
## 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414
## 2 2 5 2 5 2 2 2 5 5 5 2 2 2 5 5 5 5
## 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432
## 5 5 5 5 5 5 2 2 2 5 5 5 5 5 5 5 2 5
## 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450
## 2 2 5 5 5 5 5 2 2 2 2 2 2 5 2 2 2 2
## 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468
## 5 2 2 2 5 5 5 5 2 2 2 2 2 2 2 2 5 2
## 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504
## 2 2 2 2 5 2 2 4 4 4 4 4 4 4 4 4 4 4
## 505 506
## 4 4
##
## $centers
## 1 2 3 4 5 6 7 8
## 1 3.989644 4.556297 4.216580 3.977768 4.072917 4.089124 4.248097 4.850842
## 2 5.635860 5.015922 5.810840 6.243394 6.219170 5.797678 5.007134 5.045384
## 3 4.115845 4.367908 3.914214 4.415556 4.277226 4.521520 4.462950 4.591271
## 4 3.322873 2.419949 3.555546 3.901515 3.917887 3.259432 2.872770 3.145128
## 5 7.347551 6.917018 7.668887 8.029178 8.008552 7.594726 6.746593 6.719663
## 6 2.552963 2.191615 2.547985 2.481257 2.636215 2.126035 2.463163 3.260455
## 9 10 11 12 13 14 15 16
## 1 6.023239 4.730761 5.085827 4.546510 4.256435 5.022876 5.346690 5.064489
## 2 5.275376 5.034209 5.024306 5.008656 5.191566 4.483294 4.239085 4.500312
## 3 5.861940 4.897185 5.053174 4.724166 4.806196 5.041859 5.037049 5.123212
## 4 3.802664 3.078288 3.240106 2.958259 3.171899 2.294340 2.083575 2.330854
## 5 6.637733 6.672022 6.656885 6.725987 6.831291 6.522547 6.271019 6.514117
## 6 4.420281 3.050288 3.460627 2.800817 2.548810 2.498158 2.847417 2.519525
## 17 18 19 20 21 22 23 24
## 1 4.839017 5.494447 5.410418 5.409972 6.301722 5.559564 5.849309 6.119225
## 2 4.932293 4.118743 4.699219 4.231080 4.140866 4.107180 4.064205 4.123917
## 3 5.188031 5.159236 5.556072 5.185846 5.777782 5.080391 5.333884 5.590056
## 4 2.926372 2.006490 2.829410 2.073005 2.478201 1.996680 2.200197 2.363123
## 5 6.848068 6.136584 6.246617 6.243108 5.944153 6.154788 6.070741 6.047729
## 6 2.477710 2.972866 3.035484 2.815474 3.883785 3.047242 3.413583 3.706860
## 25 26 27 28 29 30 31 32
## 1 5.776453 5.909337 5.645826 5.792679 5.463721 5.291108 6.251560 5.790675
## 2 4.125236 4.229138 4.173627 4.154204 4.209655 4.253839 4.190000 4.132815
## 3 5.355804 5.648491 5.331440 5.423799 4.956368 4.757484 5.818922 5.290357
## 4 2.152358 2.399737 2.130704 2.313981 2.194705 2.226566 2.563711 2.185395
## 5 6.124494 5.850285 6.119300 5.848617 6.281408 6.323214 5.926407 6.111869
## 6 3.317906 3.492059 3.174430 3.407128 3.072595 2.917289 3.883866 3.341829
## 33 34 35 36 37 38 39 40
## 1 6.577978 6.113220 6.286376 5.003424 4.975828 4.672432 4.567581 2.437628
## 2 4.543962 4.083659 4.244460 4.504907 4.504214 4.851253 4.995628 6.988766
## 3 6.168152 5.623677 5.717537 4.728770 4.767845 4.847078 4.787796 5.456166
## 4 3.295133 2.297075 2.760000 2.036929 2.052171 2.484306 2.720323 5.019539
## 5 5.702765 5.899302 5.635547 6.472198 6.389736 6.751702 6.860251 8.586641
## 6 4.437132 3.655982 4.014459 2.356225 2.324457 2.085109 2.100542 3.469146
## 41 42 43 44 45 46 47 48
## 1 2.550075 4.003727 4.101822 4.105741 4.205545 4.493890 4.535762 5.127133
## 2 7.347068 6.266063 6.066725 6.010008 5.405380 5.401641 5.323536 4.969192
## 3 5.469978 4.947438 5.088076 5.079073 4.810066 5.080046 5.057556 5.071546
## 4 5.447668 4.089165 3.798403 3.739937 2.899376 2.938610 2.889613 2.590240
## 5 8.944151 7.985202 7.757987 7.727039 7.185569 7.146905 7.056687 6.715830
## 6 3.806323 2.464106 2.277093 2.245886 1.890630 2.124631 2.187826 2.870058
## 49 50 51 52 53 54 55 56
## 1 6.354068 4.788056 3.687296 3.635669 3.127248 3.365371 3.768251 2.573205
## 2 5.425549 5.185207 5.660587 5.607665 6.251113 6.093472 6.228695 8.172561
## 3 6.206363 5.152123 5.037840 4.798345 4.875844 5.098446 6.488456 6.248718
## 4 3.649819 2.729138 3.311553 3.194530 3.999725 3.806354 4.738612 6.360064
## 5 6.857506 6.903606 7.327416 7.333311 7.954340 7.750576 7.846534 9.656599
## 6 4.374890 2.484987 2.216591 2.155992 2.202966 2.192392 4.011314 4.620562
## 57 58 59 60 61 62 63 64
## 1 2.435993 2.586150 3.455264 3.762957 4.163099 4.634886 3.692547 3.384146
## 2 7.625711 8.169946 5.852530 5.357849 5.193164 4.988228 5.388941 5.749506
## 3 6.314081 6.255593 5.396948 5.291164 5.423722 5.411219 4.967901 5.080093
## 4 5.728217 6.380166 3.850185 3.257986 3.128914 3.007457 3.306577 3.782189
## 5 9.093653 9.562610 7.619436 7.182935 6.981514 6.757826 7.282673 7.580278
## 6 4.138053 4.788808 2.369128 2.238707 2.577586 3.001430 2.348247 2.423646
## 65 66 67 68 69 70 71 72
## 1 3.572872 2.365488 2.854471 3.828722 4.245479 3.861492 4.311560 4.482338
## 2 6.803815 7.202476 6.871459 5.702754 5.382884 5.506994 5.679697 5.332385
## 3 5.070095 5.920134 6.075955 5.331567 5.513971 5.232351 5.172823 5.258275
## 4 4.571372 5.363377 5.039822 3.540862 3.214330 3.286807 3.670384 3.233668
## 5 8.474576 8.706790 8.325184 7.481800 7.133931 7.317062 7.509722 7.139492
## 6 3.099810 3.755717 3.762213 2.164752 2.377335 2.067120 2.395258 2.279879
## 73 74 75 76 77 78 79 80
## 1 4.423047 4.374894 4.565665 4.571969 4.993178 4.715682 4.645174 4.703818
## 2 5.670343 5.628241 5.253844 4.504533 4.080734 4.392771 4.369389 4.613917
## 3 5.331200 5.256931 5.138078 4.739219 4.676271 4.796607 4.781211 5.037161
## 4 3.633211 3.612120 3.532216 2.554765 2.115305 2.435091 2.406743 2.698489
## 5 7.496987 7.426016 7.172186 6.550720 6.164051 6.445026 6.421550 6.621032
## 6 2.419650 2.395619 2.582255 2.208724 2.569263 2.284706 2.281898 2.322846
## 81 82 83 84 85 86 87 88
## 1 3.228722 3.647245 3.352955 3.510806 4.214393 4.216222 4.470996 4.539526
## 2 5.833680 5.345002 5.671433 5.439196 5.298921 5.378578 5.181960 5.119600
## 3 4.611003 4.507704 4.824876 4.799211 4.493640 4.250144 4.736784 4.543074
## 4 3.593223 2.978716 3.362220 3.049922 2.723815 2.845202 2.603436 2.477017
## 5 7.679009 7.247432 7.495607 7.269305 7.155841 7.257393 6.986917 6.981295
## 6 2.077143 2.128207 1.941076 1.906614 1.839655 1.960338 2.004413 2.011357
## 89 90 91 92 93 94 95 96
## 1 4.787782 4.431266 4.656816 4.754404 4.105175 4.001644 4.521147 4.347592
## 2 5.260876 5.460377 5.023131 4.997348 4.772459 5.169144 4.479851 5.437460
## 3 4.086432 3.906040 4.252949 4.267811 4.510455 4.766874 4.611235 4.143402
## 4 2.894806 3.107514 2.443156 2.419531 2.751041 3.222217 2.448424 2.937590
## 5 7.155386 7.332941 6.878036 6.862533 6.774331 7.083584 6.505216 7.188899
## 6 2.659893 2.401307 2.216247 2.322228 2.415933 2.468939 2.704529 2.182013
## 97 98 99 100 101 102 103 104
## 1 4.715695 4.926015 4.640776 4.444450 5.282563 5.152315 6.572203 5.683531
## 2 5.051112 6.412041 6.764467 5.837829 4.205067 4.305317 5.036975 3.815398
## 3 4.508034 3.931800 4.158120 3.883013 4.489959 4.495810 6.000236 5.001289
## 4 2.400338 4.464342 4.810546 3.603211 2.414290 2.493991 3.966604 1.972705
## 5 6.868421 8.203118 8.491487 7.681807 6.393702 6.492260 5.811685 5.987192
## 6 2.261612 3.692387 3.630574 2.739459 2.966666 2.843788 4.692793 3.172095
## 105 106 107 108 109 110 111 112
## 1 5.724543 6.103378 6.104999 5.770964 5.777602 5.809233 5.260548 5.174233
## 2 3.840428 3.789013 3.785737 3.801714 3.887880 3.778476 4.058712 3.895295
## 3 4.956142 5.253274 5.307506 4.990050 4.894350 5.027813 4.918935 4.183520
## 4 2.022137 2.164155 2.196416 2.023648 2.172750 2.042266 2.184470 2.238546
## 5 6.014221 5.881611 5.856170 5.944714 6.084653 5.930743 6.185964 6.069764
## 6 3.223448 3.618628 3.631351 3.259679 3.351457 3.335422 2.743048 2.868558
## 113 114 115 116 117 118 119 120
## 1 5.795991 5.778504 5.457332 5.719642 5.202323 5.373780 5.497713 5.357061
## 2 3.601847 3.596459 3.722199 3.590667 3.748787 3.744213 3.671311 3.776235
## 3 4.819193 4.774286 4.530960 4.818453 4.451377 4.589012 4.741708 4.798190
## 4 2.075218 2.106269 2.052459 2.053031 1.971584 1.988655 2.048517 2.045794
## 5 5.667421 5.680606 5.851170 5.493919 5.878062 5.869090 5.544023 5.809181
## 6 3.375466 3.392521 3.026294 3.301736 2.747953 2.915253 3.063432 2.869590
## 121 122 123 124 125 126 127 128
## 1 6.426088 6.565457 6.806695 7.305465 6.897705 6.650747 7.533959 7.099802
## 2 4.261019 4.162980 4.128457 4.247333 4.123875 4.163650 4.355189 3.161437
## 3 5.427653 5.424854 5.565025 6.069215 5.659762 5.427403 6.344535 5.999576
## 4 2.985164 2.938368 3.039074 3.475730 3.062302 2.973639 3.677551 2.962007
## 5 6.237519 6.139760 6.064278 6.001360 6.051491 6.152869 5.974765 5.445330
## 6 4.116775 4.252645 4.510005 5.058229 4.583383 4.343081 5.287399 4.735059
## 129 130 131 132 133 134 135 136
## 1 6.875876 7.154803 6.726119 6.686918 6.596530 6.821369 7.166802 6.870993
## 2 3.183589 3.174651 3.265601 3.234532 3.348169 3.162883 3.338913 3.134950
## 3 5.627423 6.120486 5.509731 5.516882 5.382470 5.779404 6.139554 5.674601
## 4 2.931268 3.021401 2.885217 2.835481 2.871725 2.806058 3.188140 2.903821
## 5 5.604137 5.425648 5.688097 5.652880 5.721190 5.497188 5.035538 5.535608
## 6 4.590233 4.791289 4.456575 4.402201 4.350142 4.479053 4.894039 4.582035
## 137 138 139 140 141 142 143 144
## 1 6.929533 6.859261 7.327005 7.036231 7.324311 8.387851 9.118028 8.148386
## 2 3.098547 3.195797 3.203702 3.121999 3.270140 4.203114 6.057048 4.491078
## 3 5.819167 5.632765 6.197877 5.809014 6.167568 7.327389 6.736697 6.601594
## 4 2.859060 2.928458 3.172007 2.969872 3.289011 4.421385 6.210476 4.719098
## 5 5.406565 5.610438 5.411571 5.483828 5.434774 5.732297 6.519423 5.705948
## 6 4.585322 4.572210 4.994596 4.721431 5.047669 6.196553 7.427033 6.200054
## 145 146 147 148 149 150 151 152
## 1 8.591160 8.420611 8.051529 8.485530 8.220414 7.812856 7.329573 7.603008
## 2 4.881773 4.956580 4.786268 4.850486 4.658853 4.298230 4.279168 4.390872
## 3 7.209721 6.881696 6.562636 7.091389 6.756783 6.281936 5.599987 6.029305
## 4 5.120965 5.221320 4.864032 5.058645 4.839584 4.425501 4.201583 4.325365
## 5 5.882920 5.395652 5.350043 5.860027 5.668932 5.539694 5.844264 5.733672
## 6 6.628290 6.634143 6.205886 6.539074 6.293195 5.851136 5.439102 5.651936
## 153 154 155 156 157 158 159 160
## 1 8.622664 7.648725 8.381752 8.889137 8.425203 6.168678 6.265700 7.165808
## 2 6.067192 4.403229 5.843835 6.536516 5.328492 5.218020 4.320602 4.558378
## 3 6.369979 6.052779 5.844207 6.703763 7.158351 4.234387 4.756091 5.368031
## 4 5.967163 4.417624 5.805631 6.566625 5.400744 4.276378 3.305864 4.351927
## 5 6.541819 5.446370 6.344801 6.334474 5.503366 6.942094 6.074535 6.178603
## 6 6.952748 5.752673 6.793409 7.450863 6.650053 4.652464 4.273222 5.383270
## 161 162 163 164 165 166 167 168
## 1 7.231434 6.383671 7.606315 7.711715 6.150603 6.283786 6.542030 6.239898
## 2 5.940257 6.088162 7.394089 7.646620 4.086328 4.360674 6.228342 4.356613
## 3 4.605274 4.398804 4.791045 4.937649 4.851661 4.926498 4.456582 5.095670
## 4 5.239834 5.208814 6.712875 6.998166 3.062315 3.452702 5.427142 3.432231
## 5 6.821006 7.745103 8.432676 8.695086 5.936568 5.663492 7.854149 5.576630
## 6 5.752473 5.237762 6.745173 6.961257 4.080738 4.385762 5.475612 4.315636
## 169 170 171 172 173 174 175 176
## 1 6.213385 6.133807 6.433082 6.312035 5.454412 4.962647 4.941506 4.309193
## 2 4.170734 4.100748 4.054129 4.025673 4.699883 4.808037 4.816591 5.496225
## 3 4.728033 4.677209 5.202442 5.015219 4.676351 4.091021 4.407110 4.153032
## 4 3.234454 3.142836 3.181914 3.077576 2.488721 2.512347 2.447970 3.326034
## 5 5.730715 5.794348 5.516106 5.715890 6.426784 6.668081 6.607389 7.275149
## 6 4.254128 4.150282 4.374444 4.230784 3.104965 2.654763 2.534686 2.342553
## 177 178 179 180 181 182 183 184
## 1 4.520962 4.655885 4.649899 4.604653 5.132343 4.814462 5.087352 5.054530
## 2 5.015596 4.976181 5.157347 5.945635 6.198433 5.587189 5.931557 5.543818
## 3 4.411335 4.095254 3.737891 3.879091 3.800673 4.190561 3.860567 4.061937
## 4 2.658053 2.616342 2.975519 3.642452 4.178165 3.143454 3.744498 3.144853
## 5 6.792977 6.830833 7.010189 7.720425 7.949502 7.318332 7.703218 7.337983
## 6 2.175426 2.365924 2.589714 2.810682 3.656914 2.701373 3.342620 2.979474
## 185 186 187 188 189 190 191 192
## 1 5.360158 4.777155 5.085672 3.049814 2.837691 2.892236 2.592917 2.487691
## 2 5.146074 5.258702 6.977902 6.090983 6.253786 6.421824 6.840347 6.509686
## 3 4.706625 4.276810 4.103340 4.369622 4.665779 4.360736 4.840840 4.814221
## 4 2.617485 2.667473 5.069928 4.220068 4.355866 4.608931 5.027524 4.620020
## 5 6.823279 6.988966 8.637738 7.764729 7.876360 8.097468 8.409214 8.135143
## 6 3.009695 2.461779 4.108449 2.947084 2.866184 3.171421 3.299804 2.939362
## 193 194 195 196 197 198 199 200
## 1 2.626002 2.323019 2.285881 3.495329 2.818135 2.825122 2.840680 2.451539
## 2 6.916387 7.350478 7.173909 8.592641 8.185358 7.919777 8.103044 7.965488
## 3 4.786382 5.315615 5.265629 5.625695 5.786297 5.784978 5.720384 6.255528
## 4 5.115810 5.261122 5.038340 6.858470 6.362933 6.100498 6.288002 6.311681
## 5 8.533158 8.873489 8.672208 10.009282 9.580192 9.208468 9.481059 9.444290
## 6 3.385297 3.423873 3.258558 5.286849 4.731058 4.582926 4.715512 4.639590
## 201 202 203 204 205 206 207
## 1 2.456624 2.615509 2.868195 3.538131 3.737246 4.677119 4.523642
## 2 7.969849 7.156198 8.271244 8.640337 8.822374 4.966944 4.612421
## 3 6.272606 5.771920 5.739854 5.855113 5.957838 4.945805 4.407545
## 4 6.329487 5.273529 6.554190 6.965429 7.174721 2.820461 2.266758
## 5 9.430213 8.587107 9.723775 10.042167 10.216692 6.837520 6.613955
## 6 4.653597 3.907488 4.866644 5.491799 5.693985 2.252863 1.996460
## 208 209 210 211 212 213 214 215
## 1 5.173003 6.173648 7.246427 6.672900 7.168488 6.358530 4.424781 5.994199
## 2 4.338029 5.951508 5.923358 5.785418 5.899356 5.931043 4.957469 5.447417
## 3 4.839209 4.297639 5.214301 4.486356 5.179802 4.522080 4.441468 6.128544
## 4 2.088170 4.501917 4.748828 4.456045 4.719108 4.521710 2.774437 3.942573
## 5 6.238923 7.056415 6.886396 6.882747 6.859678 7.017952 6.877368 6.804923
## 6 2.684155 4.559083 5.472929 4.942984 5.391736 4.669469 2.088834 4.075614
## 216 217 218 219 220 221 222 223
## 1 4.533967 6.461391 5.090497 6.968673 6.593520 6.320911 6.799719 6.130295
## 2 4.734048 5.887167 4.502200 5.707295 5.831456 6.068344 5.809474 6.083430
## 3 4.493721 4.333352 3.927803 4.474244 4.098632 3.855927 4.507648 3.793452
## 4 2.431159 4.642940 2.604000 4.619602 4.623502 4.781149 4.654611 4.752233
## 5 6.701716 6.923664 6.481639 6.712071 6.932306 7.216588 6.814993 7.226735
## 6 2.008161 4.833935 2.937343 5.253528 4.990749 4.858718 5.140070 4.702050
## 224 225 226 227 228 229 230 231
## 1 4.738585 5.386965 5.891695 5.165618 4.738937 4.794451 4.323631 4.757661
## 2 4.706017 6.297753 6.888318 5.848384 4.984604 6.583434 5.480585 4.442431
## 3 3.859178 3.856074 4.287951 3.761622 3.704484 4.327344 4.344291 4.345763
## 4 2.632171 4.857703 5.604564 4.291528 3.086868 5.025870 3.556647 2.180368
## 5 6.676587 8.061548 8.562013 7.691942 6.882629 8.267586 7.279603 6.313200
## 6 2.510992 4.285144 5.053107 3.807224 2.766869 3.866941 2.455648 2.274652
## 232 233 234 235 236 237 238 239
## 1 4.653852 5.135020 5.281054 5.952219 4.649091 6.018095 4.490512 2.932965
## 2 5.189298 6.314557 6.527581 6.182940 4.494536 6.049817 5.234441 6.100275
## 3 3.705792 3.899503 3.957570 3.819634 4.306913 3.896708 3.743105 4.916380
## 4 3.349510 4.826196 5.081229 4.795798 2.237307 4.663958 3.356024 4.006857
## 5 7.086045 8.097837 8.247294 7.256051 6.371138 7.184499 7.164135 7.763565
## 6 2.845373 4.094662 4.348568 4.582063 2.164414 4.571565 2.701412 2.261099
## 240 241 242 243 244 245 246 247
## 1 2.956836 3.179208 3.559831 3.174838 2.972611 4.361124 4.499702 3.449697
## 2 5.782692 5.644636 5.418724 5.676824 6.431679 5.293119 5.282827 5.730382
## 3 4.609977 4.554218 4.836874 4.812866 5.251961 5.595467 5.698876 5.306870
## 4 3.612081 3.523994 3.199172 3.505714 4.402069 3.307803 3.374542 3.724642
## 5 7.513168 7.401734 7.139957 7.340494 8.034419 6.985702 6.964082 7.498675
## 6 2.096391 2.294939 2.317889 2.233775 2.546385 2.843542 3.010876 2.298688
## 248 249 250 251 252 253 254 255
## 1 3.980918 3.426770 3.241080 3.321813 3.366958 3.347543 4.117622 2.565061
## 2 5.332551 5.535635 6.098106 6.038428 6.187201 6.732703 7.679236 7.519106
## 3 5.197006 5.027748 5.230422 5.313640 5.406880 5.560045 5.671848 6.347835
## 4 3.315715 3.504649 4.188133 4.105072 4.276291 4.932310 6.124355 5.596288
## 5 7.123772 7.327359 7.884487 7.824608 7.913195 8.440374 9.349847 8.962012
## 6 2.612063 2.227342 2.472500 2.394111 2.529107 3.107181 4.463447 4.036629
## 256 257 258 259 260 261 262 263
## 1 2.791995 2.848603 6.222798 5.518648 5.417650 5.174118 5.463188 5.971533
## 2 7.592589 8.192506 7.613422 6.131531 5.811603 5.956088 6.518475 7.320962
## 3 6.580192 5.707147 4.551395 3.894518 4.041152 3.826233 3.886648 4.317743
## 4 5.699971 6.403033 6.371304 4.637079 4.211517 4.374564 5.047654 6.017951
## 5 8.992176 9.653615 9.041679 7.629278 7.342798 7.487956 8.005527 8.762238
## 6 4.155292 4.824381 5.880265 4.531389 4.220084 4.147002 4.710711 5.556397
## 264 265 266 267 268 269 270 271
## 1 5.408871 5.364474 5.369503 5.311210 5.468341 4.684862 5.853082 4.111760
## 2 5.873985 6.084410 5.582297 5.691716 7.365150 6.850545 6.292985 5.143047
## 3 3.944671 3.834643 4.846831 4.038079 4.202225 3.938496 4.496815 4.877938
## 4 4.361095 4.543509 3.870860 4.139572 5.912661 5.158425 4.630825 2.685788
## 5 7.399951 7.595295 6.985984 7.167083 8.815364 8.370001 7.374525 6.914144
## 6 4.321892 4.369594 3.895689 4.141040 5.180260 4.218897 4.554789 2.091791
## 272 273 274 275 276 277 278 279
## 1 3.725629 3.921894 5.523364 4.973350 3.058289 4.960332 4.861229 3.070627
## 2 5.736385 5.155769 7.186009 7.238343 6.006291 7.213299 7.355393 5.795132
## 3 4.839362 4.286203 4.148237 4.380952 4.296659 4.210140 4.469765 4.503523
## 4 3.441028 2.665134 5.602045 5.598971 3.877658 5.612713 5.710319 3.612360
## 5 7.520064 7.064234 8.378912 8.386111 7.804129 8.364865 8.493915 7.564951
## 6 2.111547 1.993339 4.916068 4.705346 2.513014 4.779926 4.711343 2.285977
## 280 281 282 283 284 285 286 287
## 1 3.561388 4.216680 3.374750 5.571630 5.408034 2.345212 2.477254 2.927409
## 2 6.491419 7.105591 6.595419 8.219563 9.973467 7.928601 6.930874 7.445534
## 3 4.173615 3.975881 4.184062 4.440886 6.268257 5.936035 5.428512 6.501695
## 4 4.285849 5.167816 4.389736 6.563865 8.344298 6.072569 4.772330 5.471836
## 5 8.122015 8.703650 8.222715 9.258337 10.933893 9.363760 8.450626 8.767961
## 6 2.790991 3.947935 2.795306 5.515391 6.980733 4.422708 3.123714 4.079753
## 288 289 290 291 292 293 294 295
## 1 2.509960 2.585977 2.400932 2.683501 2.813661 2.686863 4.345410 4.479945
## 2 6.413629 6.249971 6.494620 6.930359 7.229010 6.875832 5.592884 5.181463
## 3 5.343188 5.207865 5.287981 5.569837 5.473392 5.650164 4.941731 4.796956
## 4 4.396113 4.214046 4.535171 5.074359 5.432490 5.006978 3.576391 3.058148
## 5 8.041222 7.905277 8.061694 8.582296 8.865102 8.509272 7.358111 6.995131
## 6 2.756566 2.716768 2.843575 3.668080 3.989047 3.608840 2.497209 2.387338
## 296 297 298 299 300 301 302 303
## 1 4.101880 4.215355 4.808009 2.456182 2.336450 2.394242 3.068423 2.941845
## 2 5.702683 5.379947 5.078750 7.315906 7.652372 7.121431 5.575441 5.929574
## 3 4.531097 4.405035 5.064135 5.938158 5.803548 5.507286 4.604003 4.762883
## 4 3.690081 3.281272 3.028002 5.478237 5.872746 5.284133 3.549854 3.969692
## 5 7.527753 7.235927 6.829083 8.722048 9.087048 8.634313 7.337638 7.605560
## 6 2.491650 2.393691 2.750925 3.788908 4.039227 3.734424 2.232083 2.376030
## 304 305 306 307 308 309 310 311
## 1 2.855273 3.436586 3.684046 3.978741 3.854584 4.998677 5.185967 5.719675
## 2 6.336183 6.106407 5.435975 5.842022 5.453779 4.466812 4.127252 4.624290
## 3 4.561650 4.196815 4.266337 4.017975 4.168573 4.202224 4.535816 5.594347
## 4 4.456508 4.073418 3.192729 3.845712 3.261918 2.344549 1.843049 2.847242
## 5 8.039455 7.868780 7.241481 7.658684 7.290961 6.546441 6.168368 6.187983
## 6 2.734380 2.774880 2.361712 3.039054 2.564437 2.631756 2.582365 3.293284
## 312 313 314 315 316 317 318 319
## 1 4.937184 5.460141 5.098426 5.028976 5.293601 5.460000 5.215271 4.838880
## 2 4.467670 4.023815 4.217072 4.259199 4.159874 4.029901 4.115912 4.245101
## 3 4.476328 4.594927 4.348331 4.178642 4.891978 4.889172 4.794276 4.262780
## 4 2.297145 1.850451 1.966400 2.079641 1.938234 1.959365 1.906122 1.969043
## 5 6.476538 6.069256 6.284457 6.344720 6.136907 5.966724 6.078177 6.307673
## 6 2.402724 2.904430 2.574041 2.606905 2.711944 2.970980 2.658070 2.293294
## 320 321 322 323 324 325 326 327
## 1 4.842685 4.449705 4.484974 4.585730 5.057642 4.334264 4.248227 4.277525
## 2 4.240360 4.799353 4.776401 4.757420 4.457601 5.018766 5.526219 5.218292
## 3 4.534455 4.506705 4.537503 4.811214 4.996492 4.590902 4.987239 4.873621
## 4 1.963279 2.473559 2.434921 2.406058 2.072533 2.775066 3.465771 3.042737
## 5 6.250142 6.816256 6.795014 6.731231 6.397846 6.994614 7.395759 7.139104
## 6 2.254385 1.961575 1.975895 2.005309 2.474246 1.958008 2.220200 2.005009
## 328 329 330 331 332 333 334 335
## 1 4.511237 4.336792 4.068778 4.134704 3.513382 3.127529 4.341878 4.382338
## 2 4.796609 5.372181 5.613018 5.389720 5.942590 6.090114 5.485716 5.424178
## 3 4.877258 5.105806 4.900165 4.957900 5.488342 5.279997 5.053555 5.093833
## 4 2.517471 3.305539 3.594198 3.303857 3.752625 3.884973 3.226203 3.160665
## 5 6.725423 7.068932 7.323299 7.087469 7.538669 7.661878 7.331467 7.267954
## 6 2.035908 2.350131 2.312741 2.230762 2.580536 2.403384 2.205917 2.203234
## 336 337 338 339 340 341 342 343
## 1 4.485929 4.729718 4.812037 4.629130 4.744764 4.837628 2.808348 4.362098
## 2 5.347455 5.032342 4.935566 5.124323 4.979263 4.865181 6.823411 5.448631
## 3 5.156767 5.112746 5.110110 5.016265 5.044939 4.987777 4.514441 4.784059
## 4 3.056405 2.647838 2.524309 2.778949 2.590159 2.417256 4.633530 3.331511
## 5 7.189428 6.898354 6.809633 7.000683 6.865025 6.777057 8.446366 7.165257
## 6 2.186727 2.223765 2.322145 2.157595 2.205404 2.271625 3.007511 2.617582
## 344 345 346 347 348 349 350 351
## 1 2.778316 2.366030 4.320697 4.466973 2.400876 2.213242 2.925825 3.077473
## 2 5.684615 6.353501 5.565696 5.487424 7.183736 7.319965 6.755452 6.526367
## 3 4.757585 4.927327 5.421207 5.492314 6.193135 5.934386 5.524385 5.618626
## 4 3.802233 4.548197 3.369890 3.310384 5.476084 5.437402 4.699873 4.404974
## 5 7.461637 8.044999 7.305607 7.150579 8.715944 8.820537 8.437225 8.219455
## 6 2.672462 2.940953 2.423474 2.543259 3.956148 3.799867 3.066109 2.914978
## 352 353 354 355 356 357 358 359
## 1 2.847020 3.254646 3.220675 4.018615 3.810904 8.663013 8.315674 8.225028
## 2 7.254755 7.319860 8.653858 7.691074 7.769223 4.621450 4.715344 4.749805
## 3 6.269580 6.763444 7.013329 7.521812 7.391746 6.500890 6.156007 6.172788
## 4 5.498901 5.585389 6.831393 6.079486 6.140023 5.907878 5.797393 5.786448
## 5 8.766898 8.789968 10.015464 9.117773 9.216143 5.456756 5.735313 5.758167
## 6 3.882617 4.035485 5.152518 4.846552 4.789845 6.995105 6.715094 6.630405
## 360 361 362 363 364 365 366 367
## 1 7.278021 7.215827 7.464845 7.699553 8.598542 8.542707 8.530898 7.799054
## 2 2.467536 2.805217 2.333627 2.769367 4.652258 6.207836 4.624219 2.863464
## 3 6.182375 6.025704 6.270247 6.585418 6.526351 6.398780 7.725367 6.760694
## 4 4.207340 4.344110 4.232657 4.378729 5.817697 6.961379 5.534146 4.394301
## 5 4.964540 5.192113 4.710339 4.977082 5.468166 7.258598 5.920246 4.657637
## 6 5.340790 5.361453 5.509807 5.682415 6.883975 7.392644 6.618984 5.752759
## 368 369 370 371 372 373 374 375
## 1 8.916963 7.876027 8.330037 8.374001 7.563523 8.532946 8.984585 9.736883
## 2 4.770107 4.939445 6.154059 6.280595 4.367286 5.917457 3.722610 4.767839
## 3 8.102215 6.662121 5.980344 5.981912 6.056314 6.266885 7.931335 8.798177
## 4 5.856186 5.544439 6.673537 6.801323 5.253873 6.587324 5.414656 6.304659
## 5 5.030692 6.544359 7.110024 7.285275 6.133890 6.680971 4.907474 5.412285
## 6 7.067897 6.260581 7.113702 7.203297 5.999323 7.191112 6.985946 7.809059
## 376 377 378 379 380 381 382 383
## 1 7.795568 7.924073 7.787937 8.275536 8.137870 12.821812 7.966471 8.185980
## 2 3.171317 2.549903 2.441746 3.030744 2.612679 9.894275 2.535543 2.500835
## 3 6.521129 6.722821 6.526672 7.136095 7.001182 12.116793 6.800516 7.087802
## 4 4.944969 4.652842 4.462833 5.083590 4.746519 11.163729 4.647065 4.555749
## 5 5.207964 4.521828 4.798660 4.679079 4.555128 9.663159 4.661952 4.529605
## 6 6.017927 6.001561 5.834679 6.394230 6.161471 11.755809 6.015300 6.112881
## 384 385 386 387 388 389 390 391
## 1 8.197628 9.349727 8.910330 9.193301 9.152365 8.907678 8.068366 7.677835
## 2 2.535837 4.207220 3.430304 3.965891 3.922688 3.476912 2.460378 2.173913
## 3 7.080988 8.494759 7.900222 8.245243 8.254546 7.913604 7.016902 6.584068
## 4 4.556169 5.912501 5.372451 5.774218 5.768068 5.325687 4.443905 4.162427
## 5 4.555957 4.593328 4.644399 4.813092 4.798740 4.585380 4.568138 4.613924
## 6 6.125147 7.394570 6.913418 7.236737 7.210917 6.892982 5.975128 5.602574
## 392 393 394 395 396 397 398 399
## 1 7.292592 8.487710 7.524344 7.763687 7.641959 7.677186 8.002371 9.767720
## 2 2.226899 2.983346 2.121055 2.265178 2.176869 2.197872 2.365545 4.881189
## 3 6.187448 7.518315 6.415998 6.699030 6.447121 6.503431 6.938797 8.851159
## 4 4.059557 4.886988 4.152838 4.352711 4.260815 4.238695 4.395922 6.683507
## 5 4.690817 4.584419 4.693783 4.560825 4.700448 4.730041 4.591516 5.349991
## 6 5.308181 6.426170 5.489378 5.726631 5.638596 5.653609 5.914268 7.989079
## 400 401 402 403 404 405 406 407
## 1 8.388325 8.804129 8.087911 7.825523 8.542363 9.552186 11.432606 9.054357
## 2 3.040490 3.471798 2.536220 2.216427 3.279024 4.910238 7.595875 3.994384
## 3 7.463474 7.776663 6.969235 6.627020 7.610294 8.716935 10.666205 8.166864
## 4 4.959465 5.497801 4.669753 4.375806 5.225734 6.673070 9.081383 5.594715
## 5 4.478721 4.713832 4.642624 4.544640 4.647198 5.160244 7.453981 4.877537
## 6 6.400821 6.900501 6.095533 5.820693 6.578583 7.840199 10.009989 7.043493
## 408 409 410 411 412 413 414
## 1 7.570396 7.942399 7.792350 10.375746 8.547474 9.825302 8.675535
## 2 2.751697 2.684071 3.512054 6.953620 4.401901 5.499975 4.000683
## 3 6.317911 6.868005 6.505072 9.624185 7.490223 8.978058 7.777474
## 4 4.354104 4.411296 4.984789 8.179599 5.753451 6.786214 5.592512
## 5 4.684219 4.359127 4.498617 6.248943 4.283169 4.939472 4.338544
## 6 5.605718 5.909119 6.091597 9.003486 6.898431 8.111524 6.840670
## 415 416 417 418 419 420 421
## 1 11.272810 9.378378 8.990931 9.186224 12.321200 8.708818 7.649405
## 2 6.897999 4.863694 4.682977 4.396530 8.940339 4.410853 2.275161
## 3 10.507271 8.404166 8.034930 8.321960 11.623093 7.789436 6.412841
## 4 8.419684 6.420781 6.131883 6.083046 10.281934 5.925977 4.353909
## 5 6.006137 4.335355 4.349290 4.150387 7.922425 4.248090 4.384831
## 6 9.673650 7.712449 7.339434 7.393740 11.103741 7.052418 5.696703
## 422 423 424 425 426 427 428 429
## 1 7.665560 7.356701 8.626415 8.441732 9.201143 8.265300 9.461813 8.241034
## 2 2.180745 2.508136 4.580701 4.667841 4.741929 4.569780 5.669296 3.716263
## 3 6.540748 6.403579 7.807594 7.858624 8.314551 7.743199 8.719330 7.368712
## 4 4.220529 4.151550 5.802264 5.736500 6.219861 5.681331 7.081609 5.224531
## 5 4.337641 4.332067 4.287058 4.467816 4.150667 4.462962 5.101391 3.944706
## 6 5.633333 5.359069 6.942560 6.733842 7.490506 6.578878 7.976041 6.437220
## 430 431 432 433 434 435 436 437
## 1 8.696259 7.931573 8.088359 7.540841 8.039230 8.323316 8.488298 8.855936
## 2 4.134503 3.794201 3.973307 3.782250 3.619313 3.721287 3.748221 4.457023
## 3 7.717744 7.082043 7.134695 6.790664 7.058495 7.351287 7.386206 7.874780
## 4 5.654346 5.081922 5.298499 4.904392 5.104864 5.332295 5.460661 5.994414
## 5 4.007002 4.145470 4.267862 4.462911 4.105578 3.959357 3.969580 4.147301
## 6 6.946892 6.173374 6.398175 5.806166 6.272329 6.545788 6.747217 7.180959
## 438 439 440 441 442 443 444 445
## 1 9.396533 9.435637 8.099988 8.384080 7.713189 7.607664 7.769558 8.369895
## 2 4.811409 4.695929 2.433801 2.937692 2.207332 2.192461 2.239337 2.871156
## 3 8.394236 8.484083 7.006532 7.350912 6.477458 6.351316 6.518089 7.288150
## 4 6.386947 6.327543 4.552303 5.079909 4.380978 4.219854 4.422982 4.931360
## 5 4.213632 4.342741 4.516857 4.509393 4.608229 4.755074 4.637509 3.918161
## 6 7.707796 7.700430 6.054304 6.446442 5.755888 5.609025 5.811139 6.418899
## 446 447 448 449 450 451 452 453
## 1 8.869383 7.692356 7.719083 7.662880 7.734779 8.619425 7.510000 7.420544
## 2 4.315338 2.223077 2.207460 2.126064 2.281549 4.610171 2.238968 2.096565
## 3 7.822260 6.503523 6.592742 6.535974 6.588139 7.643669 6.322861 6.315283
## 4 5.892565 4.313539 4.358067 4.262430 4.358030 5.922856 4.230132 4.095576
## 5 4.052052 4.377493 4.626360 4.619462 4.290924 4.436440 4.659548 4.698128
## 6 7.168870 5.714462 5.725384 5.655226 5.767098 7.013088 5.561251 5.419305
## 454 455 456 457 458 459 460 461
## 1 7.488797 8.615112 8.280416 8.613939 8.529698 7.413953 7.197961 7.441333
## 2 2.816811 4.549028 4.088681 4.450875 4.516870 2.377775 2.175860 2.581875
## 3 6.201144 7.628370 7.338629 7.770650 7.771267 6.463171 6.189770 6.350729
## 4 4.575975 5.918794 5.464389 5.749648 5.801327 4.266589 4.034746 4.354303
## 5 5.115979 4.321729 4.205509 4.252759 4.258883 4.236988 4.789795 4.385829
## 6 5.704355 7.016989 6.589745 6.913673 6.866815 5.484535 5.216250 5.566897
## 462 463 464 465 466 467 468 469
## 1 7.224008 7.129990 7.081032 6.761263 6.784776 8.163381 7.276556 7.067634
## 2 2.197525 2.207914 2.436963 2.400811 2.744376 4.323182 2.345760 2.548814
## 3 6.150612 6.113243 6.000826 5.995825 6.265945 7.375314 6.308918 6.388907
## 4 4.028929 4.046035 4.100801 3.940852 4.014623 5.454107 3.942868 4.149291
## 5 4.871657 4.859767 5.057480 4.982831 4.910892 4.372635 4.541440 4.648344
## 6 5.245377 5.175440 5.168781 4.825356 4.830816 6.492417 5.264318 5.133202
## 470 471 472 473 474 475 476 477
## 1 6.854434 6.816620 6.708850 6.594683 6.547902 7.501061 7.637258 7.206421
## 2 2.734118 2.311425 2.560930 2.499788 3.048901 2.434451 2.523020 2.223730
## 3 6.323801 5.996607 5.921288 5.766964 5.546301 6.666256 6.654666 6.151680
## 4 4.089454 3.742729 3.729722 3.771861 4.165060 4.053967 4.237950 3.945356
## 5 4.973919 4.971853 5.181002 5.161492 5.472082 4.527575 4.374497 4.862825
## 6 4.901491 4.816208 4.722700 4.659061 4.806139 5.415982 5.640980 5.208138
## 478 479 480 481 482 483 484 485
## 1 8.130462 7.407656 7.165587 6.328683 6.331364 6.329978 6.225603 6.407568
## 2 2.819877 2.149619 2.407795 2.841677 3.062382 3.305435 3.354835 3.007909
## 3 7.232034 6.384794 6.121775 5.803910 5.637316 5.575392 6.127800 6.133862
## 4 4.658359 4.056417 4.119396 3.815854 3.967622 4.139027 4.014627 3.945221
## 5 4.341229 4.588992 4.776866 5.387882 5.594453 5.786301 5.679794 5.329406
## 6 6.105211 5.382604 5.214375 4.428172 4.526780 4.616956 4.359692 4.497564
## 486 487 488 489 490 491 492 493
## 1 6.249662 6.708037 6.500512 7.746032 8.298286 8.670417 7.692851 7.224881
## 2 2.935648 2.355052 2.761430 3.339171 3.627365 4.033745 3.245201 3.298803
## 3 5.880484 6.022019 6.037518 6.749983 7.341242 7.743312 6.605526 6.225869
## 4 3.898792 3.748954 3.856913 4.067660 4.515760 4.922601 4.029950 3.853652
## 5 5.440584 4.972136 5.232320 5.468533 5.317298 5.364747 5.458144 5.624994
## 6 4.392934 4.727043 4.539202 5.655817 6.220009 6.633606 5.634401 5.208921
## 494 495 496 497 498 499 500 501
## 1 5.413752 5.273091 5.426681 6.011801 5.542505 5.414893 5.810165 5.688309
## 2 3.852454 3.986370 4.203245 3.717331 3.612856 3.666547 3.612745 3.521661
## 3 4.881230 4.787928 5.205773 5.359640 4.929451 4.699697 5.121802 4.887289
## 4 2.137355 2.335780 2.705197 2.322917 1.885295 1.925584 2.012924 1.880862
## 5 5.924585 6.035621 6.099197 5.645035 5.731891 5.820200 5.672489 5.683894
## 6 2.874711 2.815571 3.076628 3.570270 2.982151 2.867755 3.260115 3.150398
## 502 503 504 505 506
## 1 5.506603 5.730311 5.775209 5.732539 5.991382
## 2 4.294822 4.194855 4.555464 4.415418 4.255945
## 3 4.737543 4.960431 4.668497 4.734780 5.374542
## 4 2.309307 2.174338 2.764118 2.545516 2.378712
## 5 6.417801 6.310447 6.682914 6.547413 6.324684
## 6 3.056424 3.190441 3.518096 3.391335 3.469554
##
## $totss
## [1] 748534.8
##
## $withinss
## [1] 17974.37 26811.41 21548.41 22667.20 55771.90 25160.33
##
## $tot.withinss
## [1] 169933.6
##
## $betweenss
## [1] 578601.2myclust<-data.frame(km_bs2$cluster)
boston_scaled2$clust<-km_bs2$cluster
head(boston_scaled2)## crim zn indus chas nox rm
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916
## age dis rad tax ptratio black
## 1 -0.1198948 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159
## 2 0.3668034 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159
## 3 -0.2655490 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351
## 4 -0.8090878 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514
## 5 -0.5106743 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159
## 6 -0.3508100 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651
## lstat medv clust
## 1 -1.0744990 0.1595278 6
## 2 -0.4919525 -0.1014239 6
## 3 -1.2075324 1.3229375 6
## 4 -1.3601708 1.1815886 6
## 5 -1.0254866 1.4860323 6
## 6 -1.0422909 0.6705582 6lda.fit_bs2<-lda(clust~., data = boston_scaled2 )
lda.fit_bs2## Call:
## lda(clust ~ ., data = boston_scaled2)
##
## Prior probabilities of groups:
## 1 2 3 4 5 6
## 0.10079051 0.19960474 0.09486166 0.20553360 0.12845850 0.27075099
##
## Group means:
## crim zn indus chas nox rm
## 1 -0.4149170 2.55535505 -1.228758914 -0.1951310 -1.21919439 0.78676843
## 2 0.3880377 -0.48724019 1.165421314 -0.2723291 0.98659851 -0.28553884
## 3 -0.3613809 -0.09419977 -0.474086929 1.5321752 -0.12487357 1.27068222
## 4 -0.3580718 -0.46023584 -0.003188584 -0.2723291 -0.09478548 -0.35414265
## 5 1.4172264 -0.48724019 1.069802298 0.4545202 1.34622349 -0.73713928
## 6 -0.4055840 0.02149547 -0.740804469 -0.2723291 -0.79649957 0.09099544
## age dis rad tax ptratio black
## 1 -1.4488239 1.7464736 -0.7048880 -0.5692695 -0.8353442 0.34924852
## 2 0.7651453 -0.7898745 1.1388129 1.2431405 0.6932747 0.04498348
## 3 0.2307707 -0.3386056 -0.4961654 -0.7220694 -1.1226766 0.32813467
## 4 0.4093998 -0.2612071 -0.5865335 -0.4342609 0.2608189 0.19191309
## 5 0.8557425 -0.9615698 1.2885597 1.2934457 0.4142248 -1.68787016
## 6 -0.8223904 0.7053125 -0.5694290 -0.7355910 -0.2013102 0.37698635
## lstat medv
## 1 -0.9773530 0.8760790
## 2 0.6734731 -0.5987824
## 3 -0.6138415 1.4407282
## 4 0.1508360 -0.2838601
## 5 1.1961180 -0.8078336
## 6 -0.5996059 0.2092896
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3 LD4 LD5
## crim 0.04811996 -0.28556378 -0.55488255 0.49400398 0.05329096
## zn -0.13738829 -1.83004313 0.34546140 -0.26802062 -0.87758918
## indus 0.74925386 -0.10015651 0.61607026 -0.42031079 0.25109137
## chas 0.13287282 -0.13228082 -0.94523359 -0.16829634 0.04786106
## nox 1.21764057 -0.81216848 -0.12506389 0.27633410 0.13213424
## rm -0.12060003 -0.04058521 -0.02502279 -0.75468374 0.21331834
## age 0.17397462 0.34382124 -0.07430813 -0.37956005 -0.95205471
## dis -0.36273454 -0.54652248 0.11546588 0.26210162 0.59195828
## rad 0.61453519 0.40958433 0.29006265 -0.40963042 1.56473994
## tax 0.75124298 -1.03741454 0.22707980 -0.17126395 -0.61781814
## ptratio 0.36217649 -0.18603253 0.30060517 0.16017164 -0.53729844
## black -0.27542772 0.27016025 0.77143821 -0.87012879 0.23445845
## lstat 0.48988940 -0.40861927 -0.53017288 -0.23295699 -0.06758426
## medv 0.22977036 -0.57759705 -0.86635437 -0.06977308 -0.10361245
##
## Proportion of trace:
## LD1 LD2 LD3 LD4 LD5
## 0.7285 0.1498 0.0750 0.0298 0.0168lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
plot(lda.fit_bs2, dimen = 2)
lda.arrows(lda.fit_bs2, myscale = 3)
I think the dataset could be grouped more easily by a lower number of clusters, though I’m not sure what would be the minimum number. The top three most relevant variables according to our bi-plot are “zn”, “nox”, and “tax”.
Super.Bonus
Additional ways for visualising LDA:
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutmodel_predictors <- dplyr::select(train, -crime)
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crim)k_means_matpro <- kmeans(matrix_product, centers = 6, iter.max = 10, nstart = 1, trace=FALSE)
head(train)## zn indus chas nox rm age
## 70 0.04872402 -0.7385595 -0.2723291 -1.2573178 -0.5687797 -1.26381315
## 238 -0.48724019 -0.7196100 -0.2723291 -0.4115983 1.5276678 0.10746789
## 417 -0.48724019 1.0149946 -0.2723291 1.0727255 0.7078757 0.78955587
## 36 -0.48724019 -0.7545936 -0.2723291 -0.4806367 -0.5004637 -0.01331852
## 304 0.97058245 -0.7356442 -0.2723291 -1.0502028 0.9925258 -1.80735201
## 408 -0.48724019 1.0149946 -0.2723291 0.9001297 -0.9630200 1.11638970
## dis rad tax ptratio black lstat
## 70 1.2836322 -0.6373311 -0.3752120 0.2053014 0.4406159 -0.54096485
## 238 0.1676191 -0.1779443 -0.6006817 -0.4875567 0.3658034 -1.10950777
## 417 -0.9381836 1.6596029 1.5294129 0.8057784 -3.6705683 1.83963361
## 36 -0.2064589 -0.5224844 -0.7668172 0.3438730 0.4406159 -0.41633352
## 304 0.8057412 -0.2927910 -0.4701466 -1.0880337 0.3697466 -1.09130319
## 408 -1.1919222 1.6596029 1.5294129 0.8057784 -0.2692816 -0.07324727
## medv crime
## 70 -0.1775348 med_low
## 238 0.9750019 med_high
## 417 -1.6345152 high
## 36 -0.3949946 low
## 304 1.1489697 med_low
## 408 0.5835743 highmyclust <- NA
train$cl<-myclust
boston_scaled2$cl<-myclust
head(boston_scaled2)## crim zn indus chas nox rm
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916
## age dis rad tax ptratio black
## 1 -0.1198948 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159
## 2 0.3668034 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159
## 3 -0.2655490 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351
## 4 -0.8090878 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514
## 5 -0.5106743 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159
## 6 -0.3508100 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651
## lstat medv clust cl
## 1 -1.0744990 0.1595278 6 NA
## 2 -0.4919525 -0.1014239 6 NA
## 3 -1.2075324 1.3229375 6 NA
## 4 -1.3601708 1.1815886 6 NA
## 5 -1.0254866 1.4860323 6 NA
## 6 -1.0422909 0.6705582 6 NAhead(train)## zn indus chas nox rm age
## 70 0.04872402 -0.7385595 -0.2723291 -1.2573178 -0.5687797 -1.26381315
## 238 -0.48724019 -0.7196100 -0.2723291 -0.4115983 1.5276678 0.10746789
## 417 -0.48724019 1.0149946 -0.2723291 1.0727255 0.7078757 0.78955587
## 36 -0.48724019 -0.7545936 -0.2723291 -0.4806367 -0.5004637 -0.01331852
## 304 0.97058245 -0.7356442 -0.2723291 -1.0502028 0.9925258 -1.80735201
## 408 -0.48724019 1.0149946 -0.2723291 0.9001297 -0.9630200 1.11638970
## dis rad tax ptratio black lstat
## 70 1.2836322 -0.6373311 -0.3752120 0.2053014 0.4406159 -0.54096485
## 238 0.1676191 -0.1779443 -0.6006817 -0.4875567 0.3658034 -1.10950777
## 417 -0.9381836 1.6596029 1.5294129 0.8057784 -3.6705683 1.83963361
## 36 -0.2064589 -0.5224844 -0.7668172 0.3438730 0.4406159 -0.41633352
## 304 0.8057412 -0.2927910 -0.4701466 -1.0880337 0.3697466 -1.09130319
## 408 -1.1919222 1.6596029 1.5294129 0.8057784 -0.2692816 -0.07324727
## medv crime cl
## 70 -0.1775348 med_low NA
## 238 0.9750019 med_high NA
## 417 -1.6345152 high NA
## 36 -0.3949946 low NA
## 304 1.1489697 med_low NA
## 408 0.5835743 high NArownames(train)## [1] "70" "238" "417" "36" "304" "408" "294" "458" "51" "250" "64"
## [12] "438" "366" "299" "497" "298" "226" "387" "13" "345" "90" "362"
## [23] "427" "149" "177" "67" "32" "209" "144" "205" "193" "251" "175"
## [34] "293" "40" "429" "4" "436" "71" "494" "369" "390" "120" "114"
## [45] "322" "261" "239" "21" "22" "156" "213" "278" "467" "277" "219"
## [56] "96" "232" "109" "301" "463" "46" "11" "475" "343" "111" "5"
## [67] "489" "280" "129" "435" "57" "419" "389" "83" "208" "16" "340"
## [78] "376" "139" "35" "444" "56" "253" "58" "371" "470" "383" "225"
## [89] "502" "88" "296" "397" "196" "416" "228" "381" "216" "168" "182"
## [100] "78" "324" "313" "229" "191" "270" "102" "434" "135" "242" "405"
## [111] "487" "372" "218" "252" "80" "320" "97" "154" "37" "121" "126"
## [122] "10" "157" "414" "482" "6" "45" "49" "483" "200" "194" "450"
## [133] "471" "292" "485" "364" "311" "478" "379" "161" "271" "115" "55"
## [144] "140" "243" "401" "347" "431" "353" "346" "330" "316" "415" "355"
## [155] "274" "77" "100" "268" "14" "234" "183" "391" "309" "19" "30"
## [166] "443" "321" "27" "361" "409" "189" "180" "365" "287" "267" "199"
## [177] "79" "262" "224" "282" "388" "28" "256" "69" "15" "82" "202"
## [188] "60" "211" "246" "339" "396" "273" "289" "65" "456" "72" "398"
## [199] "174" "486" "236" "300" "351" "179" "370" "484" "323" "146" "89"
## [210] "124" "325" "490" "504" "241" "393" "428" "198" "360" "210" "118"
## [221] "181" "222" "447" "358" "230" "432" "125" "411" "201" "333" "297"
## [232] "169" "258" "133" "240" "254" "54" "474" "87" "53" "472" "84"
## [243] "338" "190" "476" "334" "85" "468" "286" "373" "283" "377" "266"
## [254] "112" "279" "42" "52" "159" "452" "335" "290" "151" "63" "402"
## [265] "20" "91" "328" "356" "143" "498" "136" "110" "158" "265" "423"
## [276] "185" "1" "442" "223" "86" "451" "147" "187" "479" "162" "104"
## [287] "186" "108" "359" "445" "141" "406" "12" "312" "184" "469" "263"
## [298] "50" "269" "160" "332" "477" "94" "441" "192" "107" "352" "44"
## [309] "395" "503" "412" "227" "422" "493" "501" "178" "439" "131" "173"
## [320] "459" "195" "235" "117" "465" "499" "163" "217" "244" "491" "93"
## [331] "73" "276" "29" "315" "207" "145" "155" "285" "113" "103" "172"
## [342] "68" "204" "327" "480" "275" "319" "453" "349" "284" "375" "336"
## [353] "386" "342" "318" "99" "66" "164" "440" "248" "62" "18" "488"
## [364] "426" "138" "148" "7" "410" "344" "188" "329" "212" "176" "326"
## [375] "41" "374" "245" "166" "23" "394" "384" "288" "231" "26" "367"
## [386] "255" "220" "130" "437" "317" "237" "119" "150" "492" "363" "457"
## [397] "233" "495" "116" "247" "449" "307" "8" "215"rownames(boston_scaled2)## [1] "1" "2" "3" "4" "5" "6" "7" "8" "9" "10" "11"
## [12] "12" "13" "14" "15" "16" "17" "18" "19" "20" "21" "22"
## [23] "23" "24" "25" "26" "27" "28" "29" "30" "31" "32" "33"
## [34] "34" "35" "36" "37" "38" "39" "40" "41" "42" "43" "44"
## [45] "45" "46" "47" "48" "49" "50" "51" "52" "53" "54" "55"
## [56] "56" "57" "58" "59" "60" "61" "62" "63" "64" "65" "66"
## [67] "67" "68" "69" "70" "71" "72" "73" "74" "75" "76" "77"
## [78] "78" "79" "80" "81" "82" "83" "84" "85" "86" "87" "88"
## [89] "89" "90" "91" "92" "93" "94" "95" "96" "97" "98" "99"
## [100] "100" "101" "102" "103" "104" "105" "106" "107" "108" "109" "110"
## [111] "111" "112" "113" "114" "115" "116" "117" "118" "119" "120" "121"
## [122] "122" "123" "124" "125" "126" "127" "128" "129" "130" "131" "132"
## [133] "133" "134" "135" "136" "137" "138" "139" "140" "141" "142" "143"
## [144] "144" "145" "146" "147" "148" "149" "150" "151" "152" "153" "154"
## [155] "155" "156" "157" "158" "159" "160" "161" "162" "163" "164" "165"
## [166] "166" "167" "168" "169" "170" "171" "172" "173" "174" "175" "176"
## [177] "177" "178" "179" "180" "181" "182" "183" "184" "185" "186" "187"
## [188] "188" "189" "190" "191" "192" "193" "194" "195" "196" "197" "198"
## [199] "199" "200" "201" "202" "203" "204" "205" "206" "207" "208" "209"
## [210] "210" "211" "212" "213" "214" "215" "216" "217" "218" "219" "220"
## [221] "221" "222" "223" "224" "225" "226" "227" "228" "229" "230" "231"
## [232] "232" "233" "234" "235" "236" "237" "238" "239" "240" "241" "242"
## [243] "243" "244" "245" "246" "247" "248" "249" "250" "251" "252" "253"
## [254] "254" "255" "256" "257" "258" "259" "260" "261" "262" "263" "264"
## [265] "265" "266" "267" "268" "269" "270" "271" "272" "273" "274" "275"
## [276] "276" "277" "278" "279" "280" "281" "282" "283" "284" "285" "286"
## [287] "287" "288" "289" "290" "291" "292" "293" "294" "295" "296" "297"
## [298] "298" "299" "300" "301" "302" "303" "304" "305" "306" "307" "308"
## [309] "309" "310" "311" "312" "313" "314" "315" "316" "317" "318" "319"
## [320] "320" "321" "322" "323" "324" "325" "326" "327" "328" "329" "330"
## [331] "331" "332" "333" "334" "335" "336" "337" "338" "339" "340" "341"
## [342] "342" "343" "344" "345" "346" "347" "348" "349" "350" "351" "352"
## [353] "353" "354" "355" "356" "357" "358" "359" "360" "361" "362" "363"
## [364] "364" "365" "366" "367" "368" "369" "370" "371" "372" "373" "374"
## [375] "375" "376" "377" "378" "379" "380" "381" "382" "383" "384" "385"
## [386] "386" "387" "388" "389" "390" "391" "392" "393" "394" "395" "396"
## [397] "397" "398" "399" "400" "401" "402" "403" "404" "405" "406" "407"
## [408] "408" "409" "410" "411" "412" "413" "414" "415" "416" "417" "418"
## [419] "419" "420" "421" "422" "423" "424" "425" "426" "427" "428" "429"
## [430] "430" "431" "432" "433" "434" "435" "436" "437" "438" "439" "440"
## [441] "441" "442" "443" "444" "445" "446" "447" "448" "449" "450" "451"
## [452] "452" "453" "454" "455" "456" "457" "458" "459" "460" "461" "462"
## [463] "463" "464" "465" "466" "467" "468" "469" "470" "471" "472" "473"
## [474] "474" "475" "476" "477" "478" "479" "480" "481" "482" "483" "484"
## [485] "485" "486" "487" "488" "489" "490" "491" "492" "493" "494" "495"
## [496] "496" "497" "498" "499" "500" "501" "502" "503" "504" "505" "506"train$cl <- boston_scaled2$clust[match(rownames(train), rownames(boston_scaled2))]
head(train)## zn indus chas nox rm age
## 70 0.04872402 -0.7385595 -0.2723291 -1.2573178 -0.5687797 -1.26381315
## 238 -0.48724019 -0.7196100 -0.2723291 -0.4115983 1.5276678 0.10746789
## 417 -0.48724019 1.0149946 -0.2723291 1.0727255 0.7078757 0.78955587
## 36 -0.48724019 -0.7545936 -0.2723291 -0.4806367 -0.5004637 -0.01331852
## 304 0.97058245 -0.7356442 -0.2723291 -1.0502028 0.9925258 -1.80735201
## 408 -0.48724019 1.0149946 -0.2723291 0.9001297 -0.9630200 1.11638970
## dis rad tax ptratio black lstat
## 70 1.2836322 -0.6373311 -0.3752120 0.2053014 0.4406159 -0.54096485
## 238 0.1676191 -0.1779443 -0.6006817 -0.4875567 0.3658034 -1.10950777
## 417 -0.9381836 1.6596029 1.5294129 0.8057784 -3.6705683 1.83963361
## 36 -0.2064589 -0.5224844 -0.7668172 0.3438730 0.4406159 -0.41633352
## 304 0.8057412 -0.2927910 -0.4701466 -1.0880337 0.3697466 -1.09130319
## 408 -1.1919222 1.6596029 1.5294129 0.8057784 -0.2692816 -0.07324727
## medv crime cl
## 70 -0.1775348 med_low 6
## 238 0.9750019 med_high 6
## 417 -1.6345152 high 5
## 36 -0.3949946 low 4
## 304 1.1489697 med_low 6
## 408 0.5835743 high 2nrow(train)## [1] 404plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type = "scatter3d", mode="markers", color = train$cl)options(knitr.duplicate.label = 'allow')summary(cars)## speed dist
## Min. : 4.0 Min. : 2.00
## 1st Qu.:12.0 1st Qu.: 26.00
## Median :15.0 Median : 36.00
## Mean :15.4 Mean : 42.98
## 3rd Qu.:19.0 3rd Qu.: 56.00
## Max. :25.0 Max. :120.00
Chapter 5 - Dimensionality Reduction techniques
1.Here are some correlations between the variables:
human <- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/human2.txt", sep= ",", header=TRUE, row.names = 1)
library(GGally)
library(corrplot)
library(dplyr)
library(ggplot2)
library(tidyr)
ggpairs(human)
cor(human)## Edu2.FM Labo.FM Edu.Exp Life.Exp GNI
## Edu2.FM 1.000000000 0.009564039 0.59325156 0.5760299 0.43030485
## Labo.FM 0.009564039 1.000000000 0.04732183 -0.1400125 -0.02173971
## Edu.Exp 0.593251562 0.047321827 1.00000000 0.7894392 0.62433940
## Life.Exp 0.576029853 -0.140012504 0.78943917 1.0000000 0.62666411
## GNI 0.430304846 -0.021739705 0.62433940 0.6266641 1.00000000
## Mat.Mor -0.660931770 0.240461075 -0.73570257 -0.8571684 -0.49516234
## Ado.Birth -0.529418415 0.120158862 -0.70356489 -0.7291774 -0.55656208
## Parli.F 0.078635285 0.250232608 0.20608156 0.1700863 0.08920818
## Mat.Mor Ado.Birth Parli.F
## Edu2.FM -0.6609318 -0.5294184 0.07863528
## Labo.FM 0.2404611 0.1201589 0.25023261
## Edu.Exp -0.7357026 -0.7035649 0.20608156
## Life.Exp -0.8571684 -0.7291774 0.17008631
## GNI -0.4951623 -0.5565621 0.08920818
## Mat.Mor 1.0000000 0.7586615 -0.08944000
## Ado.Birth 0.7586615 1.0000000 -0.07087810
## Parli.F -0.0894400 -0.0708781 1.00000000dim(human)## [1] 155 8str(human)## 'data.frame': 155 obs. of 8 variables:
## $ Edu2.FM : num 1.007 0.997 0.983 0.989 0.969 ...
## $ Labo.FM : num 0.891 0.819 0.825 0.884 0.829 ...
## $ Edu.Exp : num 17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
## $ Life.Exp : num 81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
## $ GNI : int 64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
## $ Mat.Mor : int 4 6 6 5 6 7 9 28 11 8 ...
## $ Ado.Birth: num 7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
## $ Parli.F : num 39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...colnames(human)## [1] "Edu2.FM" "Labo.FM" "Edu.Exp" "Life.Exp" "GNI" "Mat.Mor"
## [7] "Ado.Birth" "Parli.F"head(human)## Edu2.FM Labo.FM Edu.Exp Life.Exp GNI Mat.Mor Ado.Birth
## Norway 1.0072389 0.8908297 17.5 81.6 64992 4 7.8
## Australia 0.9968288 0.8189415 20.2 82.4 42261 6 12.1
## Switzerland 0.9834369 0.8251001 15.8 83.0 56431 6 1.9
## Denmark 0.9886128 0.8840361 18.7 80.2 44025 5 5.1
## Netherlands 0.9690608 0.8286119 17.9 81.6 45435 6 6.2
## Germany 0.9927835 0.8072289 16.5 80.9 43919 7 3.8
## Parli.F
## Norway 39.6
## Australia 30.5
## Switzerland 28.5
## Denmark 38.0
## Netherlands 36.9
## Germany 36.9The dataset created and used in this exercise is composed of eight variables and 155 observations. Out of the included variables, “GNI” and “Mat.Mor” are integer variables and the other variables are all numerical. In the following table, the information stored (by variables) is shown and elaborated
Variable - Explanation
Labo.FM - ratio of females and males in the labour force Edu.Exp - expected years of schooling Life.Exp - life expectancy at birth GNI - gross national income per capita Mat.Mor - maternal mortality ratio Ado.Birth - adolescent birth rate Parli.F - percentage of female representatives in parliament
options(knitr.duplicate.label = 'allow', debug = TRUE)
library(pander)##
## Attaching package: 'pander'## The following object is masked from 'package:GGally':
##
## wrappandoc.table(summary(human), caption = "Summary of Human data", split.table = 80)##
## -----------------------------------------------------------------
## Edu2.FM Labo.FM Edu.Exp Life.Exp
## ---------------- ---------------- --------------- ---------------
## Min. :0.1717 Min. :0.1857 Min. : 5.40 Min. :49.00
##
## 1st Qu.:0.7264 1st Qu.:0.5984 1st Qu.:11.25 1st Qu.:66.30
##
## Median :0.9375 Median :0.7535 Median :13.50 Median :74.20
##
## Mean :0.8529 Mean :0.7074 Mean :13.18 Mean :71.65
##
## 3rd Qu.:0.9968 3rd Qu.:0.8535 3rd Qu.:15.20 3rd Qu.:77.25
##
## Max. :1.4967 Max. :1.0380 Max. :20.20 Max. :83.50
## -----------------------------------------------------------------
##
## Table: Summary of Human data (continued below)
##
##
## ------------------------------------------------------------------
## GNI Mat.Mor Ado.Birth Parli.F
## ---------------- ---------------- ---------------- ---------------
## Min. : 581 Min. : 1.0 Min. : 0.60 Min. : 0.00
##
## 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65 1st Qu.:12.40
##
## Median : 12040 Median : 49.0 Median : 33.60 Median :19.30
##
## Mean : 17628 Mean : 149.1 Mean : 47.16 Mean :20.91
##
## 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95 3rd Qu.:27.95
##
## Max. :123124 Max. :1100.0 Max. :204.80 Max. :57.50
## ------------------------------------------------------------------ggpairs(human, mapping = aes(alpha = 0.3), lower = list(combo = wrap("facethist")))
The summary shows interesting observations on the variables. The adoloscent birth rate (Ado.Birth) is positively correlated (0.759) with maternal mortality ratio but negatively correlated (-0.857) with life expectancy at birth (Life.Exp). Similarly, ratio of females and males with secondary education (Edu2.FM) and expected years of schooling (Edu.Exp) are both positively correlated with life expectancy at birth (Life.Exp). On the other hand, there is very little correlation between the ratio of females and males in labour force (Labo.FM) with “Edu.Exp” and “GNI”.
PCA analysis and a biplot (in a couple different ways)
biplot(pca_human, choices = 1:2, cex=c(0.8,1), col=c(“grey40”, “deeppink2”))
In the following section, we will summarize the principal components and make a principal component analysis (PCA) plot. First, PCA is done on non-standardized data followed up by standardized data.
pca_human<-prcomp(human)
biplot(pca_human, choices = 1:2, cex=c(0.8,1), col=c("grey40", "deeppink2"))## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
sum_pca_human<-summary(pca_human)
sum_pca_human## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912
## Proportion of Variance 9.999e-01 0.0001 0.00 0.00 0.000 0.000 0.0000
## Cumulative Proportion 9.999e-01 1.0000 1.00 1.00 1.000 1.000 1.0000
## PC8
## Standard deviation 0.1591
## Proportion of Variance 0.0000
## Cumulative Proportion 1.0000sum_pca_human_var<-sum_pca_human$sdev^2
sum_pca_human_var## [1] 3.438860e+08 3.441836e+04 6.343853e+02 1.312035e+02 1.418457e+01
## [6] 2.452081e+00 3.655943e-02 2.531638e-02pca_pr <- round(100*sum_pca_human$importance[2, ], digits = 1)
pc_lab<-paste0(names(pca_pr), " (", pca_pr, "%)")
biplot(pca_human, cex = c(0.8, 1), col = c("grey40", "deeppink2"), xlab = pc_lab[1], ylab = pc_lab[2], main = "PCA plot of non-scaled human data")## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
#biplot(pca_human, choices = 1:2, cex = c(1, 1), col = c("grey40", "deeppink2"),sub = "PC1 & PC2 with non-standardised dataset")The PCA biplot above does not provide a meaningful insight to the data as it shows that a single variable, “GNI” has a dominant impact and greater weight. Moreover, “GNI” has a larger variance compared to other variables.
Next, we will scale the variables in the human data and compute principal components and plot the results.
human_std <- scale(human)
pca_human_std <- prcomp(human_std)
biplot(pca_human, choices = 1:2, cex=c(0.8,1), col=c("grey40", "deeppink2"))## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
pca_human_s<-prcomp(human, scale. = TRUE)
sum_pca_human_s<-summary(pca_human_s)
pca_pr_s <- round(100*sum_pca_human_s$importance[2, ], digits = 1)
pc_lab<-paste0(names(pca_pr_s), " (", pca_pr_s, "%)")
sum_pca_human_var_s<-sum_pca_human_s$sdev^2
sum_pca_human_var_s## [1] 4.2883701 1.2989625 0.7657100 0.6066276 0.4381862 0.2876242 0.2106805
## [8] 0.1038390biplot(pca_human_s, cex = c(0.8, 1), col = c("grey40", "deeppink2"), xlab = pc_lab[1], ylab = pc_lab[2], main = "PCA plot of scaled human data")
Here, after standardization, we can see that the plots look different and thus the results are different. The results are different after scaling because PCA is more sensitive and informative when the original features are scaled. Also, PCA assumes that features with larger variances are more important that those with smaller variances. In the non-scaled pca plot, we observed that the variables with higher values have a bigger influence as is the case with the “GNI” variable. After scaling the data, the variance between the variables is more reasonable. The first principal component (PC1) explains 53% of the variation compared to the 100% from when the data was not scaled.
Interpreting the two principal component dimensions: (1). Correlations between variables: The smaller angle between the arrows explains the greater correlation between the variables. With this assumption in mind, we can see that four of the variables, “Edu.Exp”, “Life.Exp”, “GNU” and “EDU.FM” are correlated. Out of those, “GNU” and “EDU2.FM” have the highest correlation as explained by the arrows and the angles formed by the arrows. In the same way, the variables “Parli.F” and “Labo.FM” are also correlated as are the variables “Mat.Mor” and “Ado.Birth”. In addition, the plot shows that the variables “Life.Exp” and “Ado.Birth” are the least correlated as they are furthest in the plot (indicated by the large angle between these two variables).
(2). Correlation between variables and Principal components: It is assumed that the smaller the angle between the variables and principal components, the more positively correlated the variable is. In light of the assumption, the variables “Parli.F” and “Labo.FM” are positively correlated to PC1 (i.e they are contributing the direction of PC1) whereas other variables are positively correlated to PC2 and thus directing the arrows towards PC2. Also, for PC2, “Life.Exp”, “Edu2.FM”, “GNU” and “Ado.FM” have higher weights than other variables.
We will use tea data from the FactoMineR package to practice multiple correspondence analysis (MCA). In this data, there are 300 observations and 36 variables.
library(FactoMineR)
data("tea")
str(tea)## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
## $ frequency : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...dim(tea)## [1] 300 36summary(tea)## breakfast tea.time evening lunch
## breakfast :144 Not.tea time:131 evening :103 lunch : 44
## Not.breakfast:156 tea time :169 Not.evening:197 Not.lunch:256
##
##
##
##
##
## dinner always home work
## dinner : 21 always :103 home :291 Not.work:213
## Not.dinner:279 Not.always:197 Not.home: 9 work : 87
##
##
##
##
##
## tearoom friends resto pub
## Not.tearoom:242 friends :196 Not.resto:221 Not.pub:237
## tearoom : 58 Not.friends:104 resto : 79 pub : 63
##
##
##
##
##
## Tea How sugar how
## black : 74 alone:195 No.sugar:155 tea bag :170
## Earl Grey:193 lemon: 33 sugar :145 tea bag+unpackaged: 94
## green : 33 milk : 63 unpackaged : 36
## other: 9
##
##
##
## where price age sex
## chain store :192 p_branded : 95 Min. :15.00 F:178
## chain store+tea shop: 78 p_cheap : 7 1st Qu.:23.00 M:122
## tea shop : 30 p_private label: 21 Median :32.00
## p_unknown : 12 Mean :37.05
## p_upscale : 53 3rd Qu.:48.00
## p_variable :112 Max. :90.00
##
## SPC Sport age_Q frequency
## employee :59 Not.sportsman:121 15-24:92 1/day : 95
## middle :40 sportsman :179 25-34:69 1 to 2/week: 44
## non-worker :64 35-44:40 +2/day :127
## other worker:20 45-59:61 3 to 6/week: 34
## senior :35 +60 :38
## student :70
## workman :12
## escape.exoticism spirituality healthy
## escape-exoticism :142 Not.spirituality:206 healthy :210
## Not.escape-exoticism:158 spirituality : 94 Not.healthy: 90
##
##
##
##
##
## diuretic friendliness iron.absorption
## diuretic :174 friendliness :242 iron absorption : 31
## Not.diuretic:126 Not.friendliness: 58 Not.iron absorption:269
##
##
##
##
##
## feminine sophisticated slimming
## feminine :129 Not.sophisticated: 85 No.slimming:255
## Not.feminine:171 sophisticated :215 slimming : 45
##
##
##
##
##
## exciting relaxing effect.on.health
## exciting :116 No.relaxing:113 effect on health : 66
## No.exciting:184 relaxing :187 No.effect on health:234
##
##
##
##
## library(tidyr)
library(dplyr)
keep<- c("breakfast","tea.time","friends","frequency","Tea","sugar","sex","sophisticated")
my_tea <- dplyr::select(tea, one_of(keep))
gather(my_tea) %>% ggplot(aes(value)) + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8)) + facet_wrap("key", scales = "free")## Warning: attributes are not identical across measure variables;
## they will be dropped
mca_tea <- MCA(my_tea, graph=FALSE)
summary(mca_tea, nbelements=Inf, nbind=5)##
## Call:
## MCA(X = my_tea, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.213 0.189 0.159 0.136 0.131 0.118
## % of var. 15.481 13.717 11.556 9.865 9.518 8.606
## Cumulative % of var. 15.481 29.198 40.754 50.619 60.137 68.743
## Dim.7 Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.112 0.093 0.091 0.072 0.061
## % of var. 8.150 6.766 6.644 5.254 4.444
## Cumulative % of var. 76.893 83.658 90.302 95.556 100.000
##
## Individuals (the 5 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | 0.359 0.202 0.071 | 1.116 2.201 0.686 | -0.040
## 2 | -0.198 0.061 0.023 | 0.845 1.261 0.419 | 0.349
## 3 | -0.484 0.367 0.226 | -0.243 0.105 0.057 | -0.211
## 4 | 0.779 0.951 0.499 | 0.345 0.210 0.098 | -0.071
## 5 | -0.065 0.007 0.003 | 0.816 1.176 0.480 | -0.026
## ctr cos2
## 1 0.003 0.001 |
## 2 0.255 0.071 |
## 3 0.094 0.043 |
## 4 0.011 0.004 |
## 5 0.001 0.000 |
##
## Categories
## Dim.1 ctr cos2 v.test Dim.2 ctr
## breakfast | -0.545 8.384 0.275 -9.060 | 0.576 10.563
## Not.breakfast | 0.503 7.739 0.275 9.060 | -0.532 9.750
## Not.tea time | 0.663 11.263 0.340 10.090 | 0.345 3.447
## tea time | -0.514 8.730 0.340 -10.090 | -0.268 2.672
## friends | -0.115 0.504 0.025 -2.721 | -0.375 6.083
## Not.friends | 0.216 0.950 0.025 2.721 | 0.706 11.465
## 1/day | 0.296 1.631 0.041 3.487 | 0.609 7.774
## 1 to 2/week | 1.072 9.899 0.198 7.686 | -1.161 13.109
## +2/day | -0.727 13.148 0.388 -10.775 | 0.105 0.308
## 3 to 6/week | 0.502 1.674 0.032 3.100 | -0.589 2.607
## black | -0.394 2.246 0.051 -3.896 | 0.301 1.477
## Earl Grey | 0.030 0.034 0.002 0.701 | -0.174 1.295
## green | 0.707 3.224 0.062 4.295 | 0.345 0.869
## No.sugar | -0.467 6.621 0.233 -8.352 | -0.031 0.033
## sugar | 0.499 7.078 0.233 8.352 | 0.033 0.035
## F | -0.443 6.832 0.286 -9.249 | -0.357 5.014
## M | 0.646 9.969 0.286 9.249 | 0.521 7.315
## Not.sophisticated | -0.056 0.052 0.001 -0.606 | 0.786 11.599
## sophisticated | 0.022 0.020 0.001 0.606 | -0.311 4.586
## cos2 v.test Dim.3 ctr cos2 v.test
## breakfast 0.306 9.573 | -0.244 2.256 0.055 -4.060 |
## Not.breakfast 0.306 -9.573 | 0.226 2.082 0.055 4.060 |
## Not.tea time 0.092 5.254 | 0.157 0.844 0.019 2.386 |
## tea time 0.092 -5.254 | -0.121 0.654 0.019 -2.386 |
## friends 0.265 -8.898 | -0.294 4.448 0.163 -6.983 |
## Not.friends 0.265 8.898 | 0.554 8.382 0.163 6.983 |
## 1/day 0.172 7.164 | -0.206 1.058 0.020 -2.426 |
## 1 to 2/week 0.232 -8.325 | 0.110 0.139 0.002 0.787 |
## +2/day 0.008 1.552 | 0.021 0.015 0.000 0.312 |
## 3 to 6/week 0.044 -3.642 | 0.355 1.123 0.016 2.194 |
## black 0.030 2.974 | 0.821 13.085 0.221 8.125 |
## Earl Grey 0.055 -4.047 | -0.535 14.485 0.516 -12.424 |
## green 0.015 2.098 | 1.287 14.344 0.205 7.827 |
## No.sugar 0.001 -0.552 | 0.568 13.095 0.344 10.148 |
## sugar 0.001 0.552 | -0.607 13.998 0.344 -10.148 |
## F 0.186 -7.458 | 0.027 0.035 0.001 0.569 |
## M 0.186 7.458 | -0.040 0.051 0.001 -0.569 |
## Not.sophisticated 0.244 8.545 | -0.564 7.100 0.126 -6.136 |
## sophisticated 0.244 -8.545 | 0.223 2.807 0.126 6.136 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## breakfast | 0.275 0.306 0.055 |
## tea.time | 0.340 0.092 0.019 |
## friends | 0.025 0.265 0.163 |
## frequency | 0.449 0.359 0.030 |
## Tea | 0.094 0.055 0.533 |
## sugar | 0.233 0.001 0.344 |
## sex | 0.286 0.186 0.001 |
## sophisticated | 0.001 0.244 0.126 |plot(mca_tea, invisible = c("ind"), habillage = "quali", sub = "MCA of tea dataset")
In general, the MCA plot grouped the categories that are in a way, equivalent to each other, at least to some extent. I suppose it would be better to refer to them as, “similar cathegories” (both ways, as in: both are individually similar to the other one, so they share similarity to each other). Categories such as “tea time” and “friends” are grouped together and in the same way, so are the categories such as “Not friends”" and “Not.tea time”. In other words, friends tend to spend tea time together and those who do not have tea during other times (not tea times) are not close friends. The plot also indicates that females are more social than males because they have friends, and participate in tea time. It also indicates that females do not put sugar into tea, like males do.
Chapter 6 - Analysis of longitudinal data
The datasets RATS and BPRS are used for longitudinal data analysis here.
Prior to this analysis part, the data was manipulated to long form. That is shown in the R-file “meet_and_repeat.R”
- “Analyses of Ch.8”
library(dplyr)
library(stringr)
library(ggplot2)
library(GGally)
library(tidyr)
library(lme4)## Loading required package: Matrix##
## Attaching package: 'Matrix'## The following object is masked from 'package:tidyr':
##
## expandlibrary(data.table)##
## Attaching package: 'data.table'## The following object is masked from 'package:purrr':
##
## transpose## The following objects are masked from 'package:dplyr':
##
## between, first, lastratsl <- fread("~/Documents/IODS-project/data/ratsl.csv", drop = 1)
str(ratsl)## Classes 'data.table' and 'data.frame': 176 obs. of 4 variables:
## $ ID : int 1 2 3 4 5 6 7 8 9 10 ...
## $ Group : int 1 1 1 1 1 1 1 1 2 2 ...
## $ Time : chr "WD1" "WD1" "WD1" "WD1" ...
## $ Weight: int 240 225 245 260 255 260 275 245 410 405 ...
## - attr(*, ".internal.selfref")=<externalptr># Needed corrections for variable classes
ratsl$ID <- as.factor(ratsl$ID)
ratsl$Group <- as.factor(ratsl$Group)
ratsl$Time <- as.integer(substr(ratsl$Time, 3, 4))
bprs <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/BPRS.txt", sep =" ", header = T)
rats <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt", stringsAsFactors = F, na.strings = "..")
head(ratsl)## ID Group Time Weight
## 1: 1 1 1 240
## 2: 2 1 1 225
## 3: 3 1 1 245
## 4: 4 1 1 260
## 5: 5 1 1 255
## 6: 6 1 1 260str(ratsl)## Classes 'data.table' and 'data.frame': 176 obs. of 4 variables:
## $ ID : Factor w/ 16 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ Group : Factor w/ 3 levels "1","2","3": 1 1 1 1 1 1 1 1 2 2 ...
## $ Time : int 1 1 1 1 1 1 1 1 1 1 ...
## $ Weight: int 240 225 245 260 255 260 275 245 410 405 ...
## - attr(*, ".internal.selfref")=<externalptr>summary(ratsl)## ID Group Time Weight
## 1 : 11 1:88 Min. : 1.00 Min. :225.0
## 2 : 11 2:44 1st Qu.:15.00 1st Qu.:267.0
## 3 : 11 3:44 Median :36.00 Median :344.5
## 4 : 11 Mean :33.55 Mean :384.5
## 5 : 11 3rd Qu.:50.00 3rd Qu.:511.2
## 6 : 11 Max. :64.00 Max. :628.0
## (Other):110glimpse(ratsl)## Observations: 176
## Variables: 4
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1...
## $ Time <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8...
## $ Weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, ...Graphics:
ggplot(ratsl, aes(x = Time, y = Weight, linetype = ID)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times = 4)) +
facet_grid(. ~ Group, labeller = label_both) +
theme(legend.position = "none") +
scale_y_continuous(limits = c(min(ratsl$Weight), max(ratsl$Weight)),
name = "weight") +
scale_x_continuous(name = "Time (days)")
# Standardized variable weight
ratsl <- ratsl %>%
group_by(Time) %>%
mutate(stdweight = scale(Weight)) %>%
ungroup()
# New plot
ggplot(ratsl, aes(x = Time, y = stdweight, linetype = ID)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times = 4)) +
facet_grid(. ~ Group, labeller = label_both) +
theme(legend.position = "none") +
scale_y_continuous(name = "standardized weight") +
scale_x_continuous(name = "Time (days)")
# Mean response plot
n <- ratsl$Time %>%
unique() %>%
length()
ratsls <- ratsl %>%
group_by(Group, Time) %>%
summarise(mean = mean(Weight), se = sd(Weight)/sqrt(n)) %>%
ungroup()
ggplot(ratsls, aes(x = Time, y = mean, linetype = Group, shape = Group)) +
geom_line() +
scale_linetype_manual(values = c(1,2,3)) +
geom_point(size=3) +
scale_shape_manual(values = c(1,2,3)) +
geom_errorbar(aes(ymin=mean-se, ymax=mean+se, linetype="1"), width=0.3) +
theme(legend.position = c(0.8,0.4)) +
scale_y_continuous(name = "mean(weight) +/- se(weight)") +
scale_x_continuous(name = "Time (days)")
Here the weight of the individual rats is plotted with time, in their respective diet groups. There doesn’t seem to be any significant differences in weight development possibilised by a diet group. Visually, it is very difficult to notice any differences of the response variable in the different groups.
# Boxplot
ratsl64 <- ratsl %>%
filter(Time > 1) %>%
group_by(Group, ID) %>%
summarise(mean = mean(Weight)) %>%
ungroup()
ggplot(ratsl64, aes(x = Group, y = mean)) +
geom_boxplot() +
stat_summary(fun.y = "mean", geom = "point", shape=23, size=4, fill = "white") +
scale_y_continuous(name = "mean(weight), days 1-64")
# Boxplot without the outlier
ratsl641 <- ratsl %>%
filter(Time > 1, Weight < 580) %>%
group_by(Group, ID) %>%
summarise(mean = mean(Weight)) %>%
ungroup()
ggplot(ratsl641, aes(x = Group, y = mean)) +
geom_boxplot() +
stat_summary(fun.y = "mean", geom = "point", shape=23, size=4, fill = "white") +
scale_y_continuous(name = "mean(weight), days 1-64")
rats <- fread("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt", drop = 1)
str(rats)## Classes 'data.table' and 'data.frame': 16 obs. of 13 variables:
## $ ID : int 1 2 3 4 5 6 7 8 9 10 ...
## $ Group: int 1 1 1 1 1 1 1 1 2 2 ...
## $ WD1 : int 240 225 245 260 255 260 275 245 410 405 ...
## $ WD8 : int 250 230 250 255 260 265 275 255 415 420 ...
## $ WD15 : int 255 230 250 255 255 270 260 260 425 430 ...
## $ WD22 : int 260 232 255 265 270 275 270 268 428 440 ...
## $ WD29 : int 262 240 262 265 270 275 273 270 438 448 ...
## $ WD36 : int 258 240 265 268 273 277 274 265 443 460 ...
## $ WD43 : int 266 243 267 270 274 278 276 265 442 458 ...
## $ WD44 : int 266 244 267 272 273 278 271 267 446 464 ...
## $ WD50 : int 265 238 264 274 276 284 282 273 456 475 ...
## $ WD57 : int 272 247 268 273 278 279 281 274 468 484 ...
## $ WD64 : int 278 245 269 275 280 281 284 278 478 496 ...
## - attr(*, ".internal.selfref")=<externalptr>ratsl642 <- ratsl64 %>%
mutate(baseline = rats$WD1)
fit <- lm(mean ~ baseline + Group, data = ratsl642)
anova(fit)## Analysis of Variance Table
##
## Response: mean
## Df Sum Sq Mean Sq F value Pr(>F)
## baseline 1 253625 253625 1859.8201 1.57e-14 ***
## Group 2 879 439 3.2219 0.07586 .
## Residuals 12 1636 136
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Here I’m using the “Summary measure approach”. Differences in the means of the response variable are the result intended.. The averages are composed of data from the period after treatment. “ANOVA” attemps to predict group mean weights (using simple regression) by the treatment and/or the baseline weight. It can be seen that the difference between the different diets is statistically significant at the 10% level. The result is somewhat contradictory to the purely visual approach, showinf that the visual analysis is in this case, insufficient and slightly misleading.
2 -Analyses from ch.9
BPRSL data
bprsl <- fread("~/Documents/IODS-project/data/bprsl.csv", drop = 1)
bprsl$week <- as.integer(substr(bprsl$week, 5, 5))
bprsl$treatment <- as.factor(bprsl$treatment)
bprsl$subject <- as.factor(bprsl$subject)
str(bprsl)## Classes 'data.table' and 'data.frame': 360 obs. of 4 variables:
## $ treatment: Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
## $ subject : Factor w/ 20 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ week : int 0 0 0 0 0 0 0 0 0 0 ...
## $ bprs : int 42 58 54 55 72 48 71 30 41 57 ...
## - attr(*, ".internal.selfref")=<externalptr>head(bprsl)## treatment subject week bprs
## 1: 1 1 0 42
## 2: 1 2 0 58
## 3: 1 3 0 54
## 4: 1 4 0 55
## 5: 1 5 0 72
## 6: 1 6 0 48summary(bprsl)## treatment subject week bprs
## 1:180 1 : 18 Min. :0 Min. :18.00
## 2:180 2 : 18 1st Qu.:2 1st Qu.:27.00
## 3 : 18 Median :4 Median :35.00
## 4 : 18 Mean :4 Mean :37.66
## 5 : 18 3rd Qu.:6 3rd Qu.:43.00
## 6 : 18 Max. :8 Max. :95.00
## (Other):252glimpse(bprsl)## Observations: 360
## Variables: 4
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ week <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...
## $ bprs <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, ...Now let’s move to the BPRSL data. Our dataset includes four variables: the id-variable “subject”, the time variable “week”, the variable for the treatment status “treatment” and the response variable “bprs”.
Graphical display of the data
ggplot(bprsl, aes(x = week, y = bprs, col = subject)) +
geom_line(aes(linetype = treatment)) +
scale_x_continuous(name = "Time (weeks)") +
scale_y_continuous(name = "BPRS") +
theme(legend.position = "top") 
First let us experiment with some plots. At this point the BPRS-measure appears to have gradually decreased in the 8 week prediod of the data sample, undependant of the treatment status. Before further analysis, I doubt we can come into any significant conclusions.
m <- lm(bprs ~ week + treatment, data = bprsl)
summary(m)##
## Call:
## lm(formula = bprs ~ week + treatment, data = bprsl)
##
## Residuals:
## Min 1Q Median 3Q Max
## -22.454 -8.965 -3.196 7.002 50.244
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.4539 1.3670 33.982 <2e-16 ***
## week -2.2704 0.2524 -8.995 <2e-16 ***
## treatment2 0.5722 1.3034 0.439 0.661
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.37 on 357 degrees of freedom
## Multiple R-squared: 0.1851, Adjusted R-squared: 0.1806
## F-statistic: 40.55 on 2 and 357 DF, p-value: < 2.2e-16Now I’m attempting analysis with “linear mixed models”. These include “the random intercept model”, “the random intercept and slope model” and “the interaction model”, All of the models are similar, the 2nd and 3rd ones are simply expanded versions of “the random intercept model”. First I will use the naive linear model. This does not include the repeated measures structure of the data. The model indicates that the BPRS-measure decreases over time, and that this change is statistically significant. The average for the baseline BPRS-value of the subjects of treatment group two is slightly higher (Not significantly).
Random intercept model
bprsl_m1 <- lmer(bprs ~ week + treatment + (1 | subject), data = bprsl, REML = FALSE)
summary(bprsl_m1)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (1 | subject)
## Data: bprsl
##
## AIC BIC logLik deviance df.resid
## 2748.7 2768.1 -1369.4 2738.7 355
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0481 -0.6749 -0.1361 0.4813 3.4855
##
## Random effects:
## Groups Name Variance Std.Dev.
## subject (Intercept) 47.41 6.885
## Residual 104.21 10.208
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 1.9090 24.334
## week -2.2704 0.2084 -10.896
## treatment2 0.5722 1.0761 0.532
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.437
## treatment2 -0.282 0.000This model allows different intercept terms for individual observations. This changes the standard error as well as the coefficient. The coefficients of “week” and “treatment” are the same as they were in the naive linear model, but the standard errors aren’t. Instead, the errors are smaller in this case.
Random intercept and slope model
bprsl_m2 <- lmer(bprs ~ week + treatment + (week | subject), data = bprsl, REML = FALSE)
summary(bprsl_m2)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject)
## Data: bprsl
##
## AIC BIC logLik deviance df.resid
## 2745.4 2772.6 -1365.7 2731.4 353
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.8919 -0.6194 -0.0691 0.5531 3.7976
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.8202 8.0511
## week 0.9608 0.9802 -0.51
## Residual 97.4307 9.8707
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 2.1052 22.066
## week -2.2704 0.2977 -7.626
## treatment2 0.5722 1.0405 0.550
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.582
## treatment2 -0.247 0.000Now we are enabling a different slopes for observations, changing the standard errors. The standard error on “treatment” becomes smaller, while the errors for the intercept and for “week” grow.
Next we modify the model to allow our variables to interact:
bprsl_m3 <- lmer(bprs ~ week + treatment + (week | subject) + week*treatment, data = bprsl, REML = FALSE)
summary(bprsl_m3)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject) + week * treatment
## Data: bprsl
##
## AIC BIC logLik deviance df.resid
## 2744.3 2775.4 -1364.1 2728.3 352
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0512 -0.6271 -0.0767 0.5288 3.9260
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 65.0016 8.0624
## week 0.9688 0.9843 -0.51
## Residual 96.4699 9.8219
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 47.8856 2.2522 21.262
## week -2.6283 0.3589 -7.323
## treatment2 -2.2911 1.9090 -1.200
## week:treatment2 0.7158 0.4010 1.785
##
## Correlation of Fixed Effects:
## (Intr) week trtmn2
## week -0.650
## treatment2 -0.424 0.469
## wek:trtmnt2 0.356 -0.559 -0.840For “week”, the BPRS value decreased within the 8 week period in the treatment group one by approximately -2.6 points in a week (on average). That is statistically significant (t-value: >7). The coefficient on treatment variable indicates that the subjects in treatment group two had a slightly lower average baseline BPRS value. The interaction term shows that the decrease of the BPRS value was slower in treatment group two. These additional estimates are unfortunately not statistically significant. Now I can conclude that treatment in group one seems more efficient, even if this difference isn’t significant statistically.
Likelihood ratio test
anova(bprsl_m2, bprsl_m1)## Data: bprsl
## Models:
## bprsl_m1: bprs ~ week + treatment + (1 | subject)
## bprsl_m2: bprs ~ week + treatment + (week | subject)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## bprsl_m1 5 2748.7 2768.1 -1369.4 2738.7
## bprsl_m2 7 2745.4 2772.6 -1365.7 2731.4 7.2721 2 0.02636 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(bprsl_m3, bprsl_m2)## Data: bprsl
## Models:
## bprsl_m2: bprs ~ week + treatment + (week | subject)
## bprsl_m3: bprs ~ week + treatment + (week | subject) + week * treatment
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## bprsl_m2 7 2745.4 2772.6 -1365.7 2731.4
## bprsl_m3 8 2744.3 2775.4 -1364.1 2728.3 3.1712 1 0.07495 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Now I’m attemting to check the usability of the previous models by using the chi-squared statistic. According to the 1st experiment, the random intercept model can be excluded from my test. The chi-squared statistic is small, as well as the p-value. Further investigating, we should probably pick the 3rs model. The reasoning for this comes from the larger p-value, even if the difference in the results of the remaining models was much smaller than before.
OBS VS. Fitted values
ggplot(bprsl, aes(x = week, y = bprs, col = subject)) +
geom_line(aes(linetype = treatment)) +
scale_x_continuous(name = "Time (weeks)") +
scale_y_continuous(name = "BPRSL") +
theme(legend.position = "top")
fitted <- fitted(bprsl_m3)
bprsl$fitted <- fitted
ggplot(bprsl, aes(x = week, y = fitted, col = subject)) +
geom_line(aes(linetype = treatment)) +
scale_x_continuous(name = "Time (weeks)") +
scale_y_continuous(name = "BPRS") +
theme(legend.position = "top")
There are mainly two clear observations that I’m able to make regarding this model. 1. The average intercept in treatment group two appears to be smaller than in treatment group one. 2. Slopes for the data of the 1st treatment group seem a bit steeper than those of the 2nd group.